Ouroboros: A Provably Secure Proof-of-Stake Blockchain Protocol

Abstract

We present “Ouroboros”, the first blockchain protocol based on proof of stake with rigorous security guarantees. We establish security properties for the protocol comparable to those achieved by the bitcoin blockchain protocol. As the protocol provides a “proof of stake” blockchain discipline, it offers qualitative efficiency advantages over blockchains based on proof of physical resources (e.g., proof of work). We also present a novel reward mechanism for incentivizing Proof of Stake protocols and we prove that, given this mechanism, honest behavior is an approximate Nash equilibrium, thus neutralizing attacks such as selfish mining.

Full text

Ouroboros: A Provably Secure Proof-of-Stake Blockchain Protocol
Aggelos Kiayias∗Alexander Russell†Bernardo David‡Roman Oliynykov§
August 21, 2017
Abstract
We present “Ouroboros”, the first blockchain protocol based on proof of stake with rig-
orous security guarantees. We establish security properties for the protocol comparable to
those achieved by the bitcoin blockchain protocol. As the protocol provides a “proof of stake”
blockchain discipline, it offers qualitative efficiency advantages over blockchains based on proof
of physical resources (e.g., proof of work). We also present a novel reward mechanism for in-
centivizing Proof of Stake protocols and we prove that, given this mechanism, honest behavior
is an approximate Nash equilibrium, thus neutralizing attacks such as selfish mining. We also
present initial evidence of the practicality of our protocol in real world settings by providing
experimental results on transaction confirmation and processing.
1 Introduction
A primary consideration regarding the operation of blockchain protocols based on proof of work
(PoW)—such as bitcoin [30]—is the energy required for their execution. At the time of this writ-
ing, generating a single block on the bitcoin blockchain requires a number of hashing operations
exceeding 260, which results in striking energy demands. Indeed, early calculations indicated that
the energy requirements of the protocol were comparable to that of a small country [32].
This state of affairs has motivated the investigation of alternative blockchain protocols that
would obviate the need for proof of work by substituting it with another, more energy efficient,
mechanism that can provide similar guarantees. It is important to point out that the proof of work
mechanism of bitcoin facilitates a type of randomized “leader election” process that elects one of
the miners to issue the next block. Furthermore, provided that all miners follow the protocol, this
selection is performed in a randomized fashion proportionally to the computational power of each
miner. (Deviations from the protocol may distort this proportionality as exemplified by “selfish
mining” strategies [21, 38].)
A natural alternative mechanism relies on the notion of “proof of stake” (PoS). Rather than
miners investing computational resources in order to participate in the leader election process, they
instead run a process that randomly selects one of them proportionally to the stake that each
possesses according to the current blockchain ledger.
∗University of Edinburgh and IOHK. akiayias@inf.ed.ac.uk. Work partly performed while at the National and
Kapodistrian University of Athens, supported by ERC project CODAMODA #259152. Work partly supported by
H2020 Project #653497, PANORAMIX.
†University of Connecticut. acr@cse.uconn.edu.
‡Aarhus University and IOHK, bernardo.david@iohk.io. Work partly supported by European Research Council
Starting Grant 279447.
§IOHK, roman.oliynykov@iohk.io.
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In effect, this yields a self-referential blockchain discipline: maintaining the blockchain relies on
the stakeholders themselves and assigns work to them (as well as rewards) based on the amount
of stake that each possesses as reported in the ledger. Aside from this, the discipline should make
no further “artificial” computational demands on the stakeholders. In some sense, this sounds
ideal; however, realizing such a proof-of-stake protocol appears to involve a number of definitional,
technical, and analytic challenges.
Previous work. The concept of PoS has been discussed extensively in the bitcoin forum.1Proof-
of-stake based blockchain design has been more formally studied by Bentov et al., both in conjunc-
tion with PoW [5] as well as the sole mechanism for a blockchain protocol [4]. Although Bentov
et al. showed that their protocols are secure against some classes of attacks, they do not provide
a formal model for analysing PoS based protocols or security proofs relying on precise definitions.
Heuristic proof-of-stake based blockchain protocols have been proposed (and implemented) for a
number of cryptocurrencies.2Being based on heuristic security arguments, these cryptocurrencies
have been frequently found to be deficient from the point of view of security. See [4] for a discussion
of various attacks.
It is also interesting to contrast a PoS-based blockchain protocol with a classical consensus
blockchain that relies on a fixed set of authorities (see, e.g., [17]). What distinguishes a PoS-based
blockchain from those which assume static authorities is that stake changes over time and hence
the trust assumption evolves with the system.
Another alternative to PoW is the concept of proof of space [2, 20], which has been specifically
investigated in the context of blockchain protocols [33]. In a proof of space setting, a “prover”
wishes to demonstrate the utilization of space (storage / memory); as in the case of a PoW, this
utilizes a physical resource but can be less energy demanding over time. A related concept is proof
of space-time (PoST) [28]. In all these cases, however, an expensive physical resource (either storage
or computational power) is necessary.
The PoS Design challenge. A fundamental problem for PoS-based blockchain protocols is to
simulate the leader election process. In order to achieve a fair randomized election among stake-
holders, entropy must be introduced into the system, and mechanisms to introduce entropy may be
prone to manipulation by the adversary. For instance, an adversary controlling a set of stakeholders
may attempt to simulate the protocol execution trying different sequences of stakeholder partici-
pants so that it finds a protocol continuation that favors the adversarial stakeholders. This leads
to a so called “grinding” vulnerability, where adversarial parties may use computational resources
to bias the leader election.
Our Results. We present “Ouroboros”, a provably secure proof of stake system. To the best of
our knowledge this is the first blockchain protocol of its kind with a rigorous security analysis. In
more detail, our results are as follows.
First, we provide a model that formalizes the problem of realizing a PoS-based blockchain proto-
col. The model we introduce is in the spirit of [24], focusing on persistence and liveness, two formal
properties of a robust transaction ledger. Persistence states that once a node of the system pro-
claims a certain transaction as “stable”, the remaining nodes, if queried and responding honestly,
1See “Proof of stake instead of proof of work”, Bitcoin forum thread. Posts by user “QuantumMechanic” and
others. (https://bitcointalk.org/index.php?topic=27787.0.).
2A non-exhaustive list includes NXT, Neucoin, Blackcoin, Tendermint, Bitshares.
2

will also report it as stable. Here, stability is to be understood as a predicate that will be parame-
terized by some security parameter kthat will affect the certainty with which the property holds.
(E.g., “more than kblocks deep”.) Liveness ensures that once an honestly generated transaction
has been made available for a sufficient amount of time to the network nodes, say utime steps,
it will become stable. The conjunction of liveness and persistence provides a robust transaction
ledger in the sense that honestly generated transactions are adopted and become immutable. Our
model is suitably amended to facilitate PoS-based dynamics.
Second, we describe a novel blockchain protocol based on PoS. Our protocol assumes that parties
can freely create accounts and receive and make payments, and that stake shifts over time. We
utilize a (very simple) secure multiparty implementation of a coin-flipping protocol to produce the
randomness for the leader election process. This distinguishes our approach (and prevents so called
“grinding attacks”) from other previous solutions that either defined such values deterministically
based on the current state of the blockchain or used collective coin flipping as a way to introduce
entropy [4]. Also, unique to our approach is the fact that the system ignores round-to-round
stake modifications. Instead, a snapshot of the current set of stakeholders is taken in regular
intervals called epochs; in each such interval a secure multiparty computation takes place utilizing
the blockchain itself as the broadcast channel. Specifically, in each epoch a set of randomly selected
stakeholders form a committee which is then responsible for executing the coin-flipping protocol.
The outcome of the protocol determines the set of next stakeholders to execute the protocol in the
next epoch as well as the outcomes of all leader elections for the epoch.
Third, we provide a set of formal arguments establishing that no adversary can break persistence
and liveness. Our protocol is secure under a number of plausible assumptions: (1) the network
is synchronous in the sense that an upper bound can be determined during which any honest
stakeholder is able to communicate with any other stakeholder, (2) a number of stakeholders drawn
from the honest majority is available as needed to participate in each epoch, (3) the stakeholders
do not remain offline for long periods of time, (4) the adaptivity of corruptions is subject to a small
delay that is measured in rounds linear in the security parameter (or alternatively, the players
have access to a sender-anonymous broadcast channel). At the core of our security arguments is a
probabilistic argument regarding a combinatorial notion of “forkable strings” which we formulate,
prove and also verify experimentally. In our analysis we also distinguish covert attacks, a special
class of general forking attacks. “Covertness” here is interpreted in the spirit of covert adversaries
against secure multiparty computation protocols, cf. [3], where the adversary wishes to break the
protocol but prefers not to be caught doing so. We show that covertly forkable strings are a
subclass of the forkable strings with much smaller density; this permits us to provide two distinct
security arguments that achieve different trade-offs in terms of efficiency and security guarantees.
Our forkable string analysis is a natural and fairly general tool that can be applied as part of a
security argument the PoS setting.
Fourth, we turn our attention to the incentive structure of the protocol. We present a novel
reward mechanism for incentivizing the participants to the system which we prove to be an (ap-
proximate) Nash equilibrium. In this way, attacks like block withholding and selfish-mining [21, 38]
are mitigated by our design. The core idea behind the reward mechanism is to provide positive
payoff for those protocol actions that cannot be stifled by a coalition of parties that diverges from
the protocol. In this way, it is possible to show that, under plausible assumptions, namely that
certain protocol execution costs are small, following the protocol faithfully is an equilibrium when
all players are rational.
Fifth, we introduce a stake delegation mechanism that can be seamlessly added to our blockchain
protocol. Delegation is particularly useful in our context as we would like to allow our protocol
to scale even in a setting where the set of stakeholders is highly fragmented. In such cases, the
3

delegation mechanism can enable stakeholders to delegate their “voting rights”, i.e., the right
of participating in the committees running the leader selection protocol in each epoch. As in
liquid democracy, (a.k.a. delegative democracy [23]), stakeholders have the ability to revoke their
delegative appointment when they wish independently of each other.
Given our model and protocol description we also explore how various attacks considered in
practice can be addressed within our framework. Specifically, we discuss double spending attacks,
transaction denial attacks, 51% attacks, nothing-at-stake, desynchronization attacks and others.
Finally, we present evidence regarding the efficiency of our design. First we consider double spending
attacks. For illustrative purposes, we perform a comparison with Nakamoto’s analysis for bitcoin
regarding transaction confirmation time with assurance 99.9%. Against covert adversaries, the
transaction confirmation time is from 10 to 16 times faster than that of bitcoin, depending on the
adversarial hashing power; for general adversaries confirmation time is from 5 to 10 times faster.
Moreover, our concrete analysis of double-spending attacks relies on our combinatorial analysis of
forkable and covertly forkable strings and applies to a much broader class of adversarial behavior
than Nakamoto’s more simplified analysis.3We then survey our prototype implementation and
report on benchmark experiments run in the Amazon cloud that showcase the power of our proof
of stake blockchain protocol in terms of performance.
Related Work. In parallel to the development of Ouroboros, a number of other protocols were
developed targeting various positions in the design space of distributed ledgers based on PoS.
Sleepy consensus [6] considers a fixed stakeholder distribution (i.e., stake does not evolve over
time) and targets a “mixed” corruption setting, where the adversary is allowed to be adaptive as
well as perform fail-stop and recover corruptions in addition to Byzantine faults. It is actually
straightforward to extend our analysis in this mixed corruption setting, cf. Remark 2; nevertheless,
the resulting security can be argued only in the “corruptions with delay” setting, and thus is
not fully adaptive. Snow White [7] addresses an evolving stakeholder distribution and uses a
corruption delay mechanism similar to ours for arguing security. Nevertheless, contrary to our
protocol, the Snow White design is susceptible to a “grinding” type of attack that can bias high
probability events in favor of the adversary. While this does not hurt security asymptotically, it
prevents a concrete parameterisation that does not take into account adversarial computing power.
Algorand [27] provides a distributed ledger following a Byzantine agreement per block approach
that can withstand adaptive corruptions. Given that agreement needs to be reached for each block,
such protocols will produce blocks at a rate substantially slower than a PoS blockchain (where the
slow down matches the expected length of the execution of the Byzantine agreement protocol) but
they are free of forks. In this respect, despite the existence of forks, blockchain protocols exhibit
the flexibility of permitting the clients to set the level of risk that they are willing to undertake,
allowing low risk profile clients to enjoy faster processing times in the optimistic sense. Finally,
Fruitchain [36] provides a reward mechanism and an approximate Nash equilibrium proof for a
PoW-based blockchain. We use a similar reward mechanism at the blockchain level, nevertheless
our underlying mechanics are different since we have to operate in a PoS setting. The core of the
idea is to provide a PoS analogue of “endorsing” inputs in a fair proportion using the same logic as
the PoW-based byzantine agreement protocol for honest majority from [24].
3Nakamoto’s simplifications are pointed out in [24]: the analysis considers only the setting where a block with-
holding attacker acts without interaction as opposed to a more general attacker that, for instance, tries strategically
to split the honest parties in more than one chains during the course of the double spending attack.
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Paper overview. We lay out the basic model in Sec. 2. To simplify the analysis of our protocol,
we present it in four stages that are outlined in Sec. 3. In short, in Sec. 4 we describe and analyze
the protocol in the static setting; we then transition to the dynamic setting in Sec. 5. Our incentive
mechanism and the equilibrium argument are presented in Sec. 7. We then present the protocol
enhancement with anonymous channels in Sec. 6 and with a delegation mechanism in Sec. 8.
Following this, in Sec. 9 we discuss the resilience of the protocol under various particular attacks
of interest. In Sec. 10 we discuss transaction confirmation times as well as general performance
results obtained from a prototype implementation running in the Amazon cloud.
2 Model
Time, slots, and synchrony. We consider a setting where time is divided into discrete units
called slots. A ledger, described in more detail below, associates with each time slot (at most) one
ledger block. Players are equipped with (roughly synchronized) clocks that indicate the current
slot. This will permit them to carry out a distributed protocol intending to collectively assign a
block to this current slot. In general, each slot slris indexed by an integer r∈ {1,2, . . .}, and we
assume that the real time window that corresponds to each slot has the following properties.
•The current slot is determined by a publicly-known and monotonically increasing function of
current time.
•Each player has access to the current time. Any discrepancies between parties’ local time are
insignificant in comparison with the length of time represented by a slot.
•The length of the time window that corresponds to a slot is sufficient to guarantee that
any message transmitted by an honest party at the beginning of the time window will be
received by any other honest party by the end of that time window (even accounting for
small inconsistencies in parties’ local clocks). In particular, while network delays may occur,
they never exceed the slot time window.
Transaction Ledger Properties. A protocol Π implements a robust transaction ledger provided
that the ledger that Π maintains is divided into “blocks” (assigned to time slots) that determine the
order with which transactions are incorporated in the ledger. It should also satisfy the following
two properties.
•Persistence. Once a node of the system proclaims a certain transaction tx as stable, the
remaining nodes, if queried, will either report tx in the same position in the ledger or will not
report as stable any transaction in conflict to tx. Here the notion of stability is a predicate
that is parameterized by a security parameter k; specifically, a transaction is declared stable
if and only if it is in a block that is more than kblocks deep in the ledger.
•Liveness. If all honest nodes in the system attempt to include a certain transaction, then
after the passing of time corresponding to uslots (called the transaction confirmation time),
all nodes, if queried and responding honestly, will report the transaction as stable.
In [26, 35] it was shown that persistence and liveness can be derived from the following three
elementary properties provided that protocol Π derives the ledger from a data structure in the form
of a blockchain.
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•Common Prefix (CP); with parameters k∈N.The chains C1,C2possessed by two
honest parties at the onset of the slots sl1< sl2are such that Cdk
1 C2, where Cdk
1denotes
the chain obtained by removing the last kblocks from C1, and denotes the prefix relation.
•Chain Quality (CQ); with parameters µ∈(0,1] and `∈N.Consider any portion of
length at least `of the chain possessed by an honest party at the onset of a round; the ratio of
blocks originating from the adversary is at most 1−µ. We call µthe chain quality coefficient.
•Chain Growth (CG); with parameters τ∈(0,1], s ∈N.Consider the chains C1,C2
possessed by two honest parties at the onset of two slots sl1, sl2with sl2at least sslots ahead
of sl1. Then it holds that len(C2)−len(C1)≥τ·s. We call τthe speed coefficient.
Some remarks are in place. Regarding common prefix, we capture a strong notion of common
prefix, cf. [26]. Regarding chain quality, µ, as a function of the ratio of adversarial parties, satisfies
µ(α)≥αfor protocols of interest. In an ideal setting, µwould be 1−α: in this case, the percentage
of malicious blocks in any sufficiently long chain segment is proportional to the cumulative stake
of a set of (malicious) stakeholders.
It is worth noting that for bitcoin we have µ(α) = (1 −2α)/(1 −α), and this bound is in fact
tight—see [24], which argues this guarantee on chain quality. The same will hold true for our
protocol construction. As we will show, this will still be sufficient for our incentive mechanism to
work properly.
Finally chain growth concerns the rate at which the chain grows (for honest parties). As in
the case of bitcoin, the longest chain plays a preferred role in our protocol; this provides an easy
guarantee of chain growth.
Security Model. We adopt the model introduced by [24] for analysing security of blockchain
protocols enhanced with an ideal functionality F. We denote by VIEWP,F
Π,A,Z(λ) the view of party
Pafter the execution of protocol Π with adversary A, environment Z, security parameter κand
access to ideal functionality F. Similarly we denote by EXECP,F
Π,A,Z(λ) the output of Z.
We note that multiple different “functionalities” will be encompassed by F. Contrary to [24],
our analysis is in the “standard model”, and without a random oracle functionality. The first inter-
faces we incorporate in the ideal functionality used in the protocol are the “diffuse” and “key and
transaction” functionality, denoted FD+KT and described below. Note that the diffuse functionality
is also the mechanism via which we will obtain the synchronization of the protocol.
Diffuse functionality. The diffuse functionality maintains an incoming string for each party Ui
that participates. A party, if activated, is allowed at any moment to fetch the contents of its
incoming string; one may think of this as a mailbox. Additionally, parties can instruct the
functionality to diffuse a message, in which case the message will be appended to each party’s
incoming string. The functionality maintains rounds (slots) and all parties are allowed to
diffuse once in a round. Rounds do not advance unless all parties have diffused a message.
The adversary, when activated, may also interact with the functionality and is allowed to read
all inboxes and all diffuse requests and deliver messages to the inboxes in any order it prefers.
At the end of the round, the functionality will ensure that all inboxes contain all messages
that have been diffused (but not necessarily in the same order they have been requested to be
diffused). The current slot index may be requested at any time by any party. If a stakeholder
does not fetch in a certain slot the messages written to its incoming string, they are flushed.
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Key and Transaction functionality. The key registration functionality is initialized with n
users, U1, . . . , Unand their respective stake s1, . . . , sn; given such initialization, the func-
tionality will consult with the adversary and will accept a (possibly empty) sequence of
(Corrupt, U) messages and mark the corresponding users Uas corrupt. For the corrupt users
without a public-key registered the functionality will allow the adversary to set their public-
keys while for honest users the functionality will sample public/secret-key pairs and record
them based on a digital signature algorithm. Public-keys of corrupt users will be marked as
such. Subsequently, any sequence of the following actions may take place: (i) A user may
request to retrieve its public and secret-key whereupon the functionality will return it to the
user. (ii) The whole directory of public-keys may be required whereupon the functionality
will return it to the requesting user. (iii) A new user may be requested to be created by a
message (Create, U, C) from the environment, in which case the functionality will follow the
same procedure as before: it will consult the adversary regarding the corruption status of U
and will set its public and possibly secret-key depending on the corruption status; moreover
it will store Cas the suggested initial state. The functionality will return the public-key
back to the environment upon successful completion of this interaction. (iv) An existing user
may be requested to be corrupted by the adversary via a message (Corrupt, U). A user can
only be corrupted after a delay of Dslots; specifically, after a corruption request is regis-
tered the secret-key will be released after Dslots have passed according to the round counter
maintained in the Diffuse component of the functionality.
Given the above we will assume that the execution of the protocol is with respect to a functional-
ity Fthat is incorporating the above two functionalities as well as possibly additional functionalities
to be explained below. Note that a corrupted stakeholder Uwill relinquish its entire state to A;
from this point on, the adversary will be activated in place of the stakeholder U. Beyond any
restrictions imposed by F, the adversary can only corrupt a stakeholder if it is given permission
by the environment Zrunning the protocol execution. The permission is in the form of a mes-
sage (Corrupt, U) which is provided to the adversary by the environment. In summary, regarding
activations we have the following.
•At each slot slj, the environment Zis allowed to activate any subset of stakeholders it
wishes. Each one of them will possibly produce messages that are to be transmitted to other
stakeholders.
•The adversary is activated at least as the last entity in each slj, (as well as during all adver-
sarial party activations).
It is easy to see that the model above confers such sweeping power on the adversary that one
cannot establish any significant guarantees on protocols of interest. It is thus important to restrict
the environment suitably (taking into account the details of the protocol) so that we may be able
to argue security. With foresight, the restrictions we will impose on the environment are as follows.
Restrictions imposed on the environment. The environment, which is responsible for acti-
vating the honest parties in each round, will be subject to the following constraints regarding the
activation of the honest parties running the protocol.
•In each slot there will be at least one honest activated party.
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•There will be a parameter k∈Zthat will signify the maximum number of slots that an honest
shareholder can be offline. In case an honest stakeholder is spawned after the beginning of
the protocol via (Create, U, C) its initialization chain Cprovided by the environment should
match an honest parties’ chain which was active in the previous slot.
•In each slot slr, and for each active stakeholder Ujthere will be a set Sj(r) of public-keys
and stake pairs of the form (vki, si)∈ {0,1}∗×N, for j= 1, . . . , nrwhere nris the number
of users introduced up to that slot that will represent who are the active participants in the
view of Uj. Public-keys will be marked as “corrupted” if the corresponding stakeholder has
been corrupted. We will say the adversary is restricted to less than 50% relative stake if it
holds that the total stake of the corrupted keys divided by the total stake Pisiis less than
50% in all possible Sj(r). In case the above is violated an event Bad1/2becomes true for the
given execution.
We note that the offline restriction stated above is very conservative and our protocol can
tolerate much longer offline times depending on the way the course of the execution proceeds;
nevertheless, for the sake of simplicity, we use the above restriction. Finally, we note that in all our
proofs, whenever we say that a property Qholds with high probability over all executions, we will
in fact argue that Q∨Bad1/2holds with high probability over all executions. This captures the fact
that we exclude environments and adversaries that trigger Bad1/2with non-negligible probability.
3 Our Protocol: Overview
We first provide a general overview of our protocol design approach. The protocol’s specifics depend
on a number of parameters as follows: (i) kis the number of blocks a certain message should have
“on top of it” in order to become part of the immutable history of the ledger, (ii) is the advantage
in terms of stake of the honest stakeholders against the adversarial ones; (iii) Dis the corruption
delay that is imposed on the adversary, i.e., an honest stakeholder will be corrupted after Dslots
when a corrupt message is delivered by the adversary during an execution; (iv) Lis the lifetime of
the system, measured in slots; (v) Ris the length of an epoch, measured in slots.
We present our protocol description in four stages successively improving the adversarial model
it can withstand. In all stages an “ideal functionality” FD,F
LS is available to the participants. The
functionality captures the resources that are available to the parties as preconditions for the secure
operation of the protocol (e.g., the genesis block will be specified by FD,F
LS ).
Stage 1: Static stake; D=L.In the first stage, the trust assumption is static and remains
with the initial set of stakeholders. There is an initial stake distribution which is hardcoded
into the genesis block that includes the public-keys of the stakeholders, {(vki, si)}n
i=1. Based on
our restrictions to the environment, honest majority with advantage is assumed among those
initial stakeholders. Specifically, the environment initially will allow the corruption of a number
of stakeholders whose relative stake represents 1−
2for some  > 0. The environment allows party
corruption by providing tokens of the form (Corrupt, U ) to the adversary; note that due to the
corruption delay imposed in this first stage any further corruptions will be against parties that
have no stake initially and hence the corruption model is akin to “static corruption.” FD,F
LS will
subsequently sample ρwhich will seed a “weighted by stake” stakeholder sampling and in this way
lead to the election of a subset of mkeys vki1,...,vkimto form the committee that will possess
honest majority with overwhelming probability in m, (this uses the fact that the relative stake
possessed by malicious parties is 1−
2; a linear dependency of mto −2will be imposed at this
8

stage). In more detail, the committee will be selected implicitly by appointing a stakeholder with
probability proportional to its stake to each one of the Lslots. Subsequently, stakeholders will
issue blocks following the schedule that is determined by the slot assignment. The longest chain
rule will be applied and it will be possible for the adversary to fork the blockchain views of the
honest parties. Nevertheless, we will prove with a Markov chain argument that the probability
that a fork can be maintained over a sequence of nslots drops exponentially with at least √n, cf.
Theorem 4.13 against general adversaries. An even more favorable analysis can be made against
covert adversaries, i.e., adversaries that prefer to remain “under the radar” cf. Theorem 4.23.
Stage 2: Dynamic state with a beacon, epoch period of Rslots, D=RL.The central
idea for the extension of the lifetime of the above protocol is to consider the sequential composition
of several invocations of it. We detail a way to do that, under the assumption that a trusted beacon
emits a uniformly random string in regular intervals. More specifically, the beacon, during slots
{j·R+ 1,...,(j+ 1)R}, reveals the j-th random string that seeds the leader election function. The
critical difference compared to the static state protocol is that the stake distribution is allowed to
change and is drawn from the blockchain itself. This means that at a certain slot sl that belongs
to the j-th epoch (with j≥2), the stake distribution that is used is the one reported in the most
recent block with time stamp less than j·R−2k.
Regarding the evolving stake distribution, transactions will be continuously generated and trans-
ferred between stakeholders via the environment and players will incorporate posted transactions in
the blockchain based ledgers that they maintain. In order to accomodate the new accounts that are
being created, the FD,F
LS functionality enables a new (vk,sk) to be created on demand and assigned
to a new party Ui. Specifically, the environment can create new parties who will interact with FD,F
LS
for their public/secret-key in this way treating it as a trusted component that maintains the secret
of their wallet. Note that the adversary can interfere with the creation of a new party, corrupt it,
and supply its own (adversarially created) public-key instead. As before, the environment, may
request transactions between accounts from stakeholders and it can also generate transactions in
collaboration with the adversary on behalf of the corrupted accounts. Recall that our assumption
is that at any slot, in the view of any honest player, the stakeholder distribution satisfies honest
majority with advantage (note that different honest players might perceive a different stakeholder
distribution in a certain slot). Furthermore, the stake can shift by at most σstatistical distance
over a certain number of slots. The statistical distance here will be measured considering the un-
derlying distribution to be the weighted-by-stake sampler and how it changes over the specified
time interval. The security proof can be seen as an induction in the number of epochs L/R with
the base case supplied by the proof of the static stake protocol. In the end we will argue that in this
setting, a 1−
2−σbound in adversarial stake is sufficient for security of a single draw (and observe
that the size of committee, m, now should be selected to overcome also an additive term of size
ln(L/R) given that the lifetime of the systems includes such a number of successive epochs). The
corruption delay remains at D=Rwhich can be selected arbitrarily smaller than L, thus enabling
the adversary to perform adaptive corruptions as long as this is not instantaneous.
Stage 3: Dynamic state without a beacon, epoch period of Rslots, R= Θ(k)and delay
D∈(R, 2R)L.In the third stage, we remove the dependency to the beacon, by introducing
a secure multiparty protocol with “guaranteed output delivery” that simulates it. In this way, we
can obtain the long-livedness of the protocol as described in the stage 2 design but only under
the assumption of the stage 1 design, i.e., the mere availability of an initial random string and
an initial stakeholder distribution with honest majority. The core idea is the following: given we
9

guarantee that an honest majority among elected stakeholders will hold with very high probability,
we can further use this elected set as participants to an instance of a secure multiparty computation
(MPC) protocol. This will require the choice of the length of the epoch to be sufficient so that it
can accommodate a run of the MPC protocol. From a security point of view, the main difference
with the previous case, is that the output of the beacon will become known to the adversary before
it may become known to the honest parties. Nevertheless, we will prove that the honest parties
will also inevitably learn it after a short number of slots. To account for the fact that the adversary
gets this headstart (which it may exploit by performing adaptive corruptions) we increase the wait
time for corruption from Rto a suitable value in (R, 2R) that negates this advantage and depends
on the secure MPC design. A feature of this stage from a cryptographic design perspective is the
use of the ledger itself for the simulation of a reliable broadcast that supports the MPC protocol.
Stage 4: Input endorsers, stakeholder delegates, anonymous communication. In the
final stage of our design, we augment the protocol with two new roles for the entities that are
running the protocol and consider the benefits of anonymous communication. Input-endorsers
create a second layer of transaction endorsing prior to block inclusion. This mechanism enables
the protocol to withstand deviations such as selfish mining and enables us to show that honest
behaviour is an approximate Nash equilibrium under reasonable assumptions regarding the costs
of running the protocol. Note that input-endorsers are assigned to slots in the same way that
slot leaders are, and inputs included in blocks are only acceptable if they are endorsed by an
eligible input-endorser. Second, the delegation feature allows stakeholders to transfer committee
participation to selected delegates that assume the responsibility of the stakeholders in running
the protocol (including participation to the MPC and issuance of blocks). Delegation naturally
gives rise to “stake pools” that can act in the same way as mining pools in bitcoin. Finally, we
observe that by including an anonymous communication layer we can remove the corruption delay
requirement that is imposed in our analysis. This is done at the expense of increasing the online
time requirements for the honest parties.4
4 Our Protocol: Static State
4.1 Basic Concepts and Protocol Description
We begin by describing the blockchain protocol πSPoS in the “static stake” setting, where leaders
are assigned to blockchain slots with probability proportional to their (fixed) initial stake which
will be the effective stake distribution throughout the execution. To simplify our presentation, we
abstract this leader selection process, treating it simply as an “ideal functionality” that faithfully
carries out the process of randomly assigning stakeholders to slots. In the following section, we
explain how to instantiate this functionality with a specific secure computation.
We remark that—even with an ideal leader assignment process—analyzing the standard “longest
chain” preference rule in our PoS setting appears to require significant new ideas. The challenge
arises because large collections of slots (epochs, as described above) are assigned to stakeholders at
once; while this has favorable properties from an efficiency (and incentive) perspective, it furnishes
the adversary a novel means of attack. Specifically, an adversary in control of a certain population
of stakeholders can, at the beginning of an epoch, choose when standard “chain update” broadcast
messages are delivered to honest parties with full knowledge of future assignments of slots to
stakeholders. In contrast, adversaries in typical PoW settings are constrained to make such decisions
4In follow-up work we show how the same can be achieved efficiently, see [18].
10

in an online fashion. We remark that this can have a dramatic effect on the ability of an adversary
to produce alternate chains; see the discussion on “forkable strings” below for detailed discussion.
In the static stake case, we assume that a fixed collection of nstakeholders U1, . . . , Uninteract
throughout the protocol. Stakeholder Uipossesses sistake before the protocol starts. For each
stakeholder Uia verification and signing key pair (vki,ski) for a prescribed signature scheme is
generated; we assume without loss of generality that the verification keys vk1, . . . are known by all
stakeholders. Before describing the protocol, we establish basic definitions following the notation
of [24].
Definition 4.1 (Genesis Block).The genesis block B0contains the list of stakeholders identified
by their public-keys, their respective stakes (vk1, s1),...,(vkn, sn)and auxiliary information ρ.
With foresight we note that the auxiliary information ρwill be used to seed the slot leader
election process.
Definition 4.2 (State).Astate is a string st ∈ {0,1}λ.
Definition 4.3 (Block).Ablock Bgenerated at a slot sli∈ {sl1, . . . , slR}contains the current
state st ∈ {0,1}λ, data d∈ {0,1}∗, the slot number sliand a signature σ=Signski(st, d, sli)
computed under skicorresponding to the stakeholder Uigenerating the block.
Definition 4.4 (Blockchain).Ablockchain (or simply chain) relative to the genesis block B0is a
sequence of blocks B1, . . . , Bnassociated with a strictly increasing sequence of slots for which the
state stiof Biis equal to H(Bi−1), where His a prescribed collision-resistant hash function. The
length of a chain len(C) = nis its number of blocks. The block Bnis the head of the chain, denoted
head(C). We treat the empty string εas a legal chain and by convention set head(ε) = ε.
Let Cbe a chain of length nand kbe any non-negative integer. We denote by Cdkthe chain
resulting from removal of the krightmost blocks of C. If k≥len(C) we define Cdk=ε. We let
C1 C2indicate that the chain C1is a prefix of the chain C2.
Definition 4.5 (Epoch).An epoch is a set of Radjacent slots S={sl1, . . . , slR}.
(The value Ris a parameter of the protocol we analyze in this section.)
Definition 4.6 (Adversarial Stake Ratio).Let UAbe the set of stakeholders controlled by an
adversary A. Then the adversarial stake ratio is defined as
α=Pj∈UAsj
Pn
i=1 si
,
where nis the total number of stakeholders and siis stakeholder Ui’s stake.
Slot Leader Selection. In the protocol described in this section, for each 0 < j ≤R, a slot
leader Ejis determined who has the (sole) right to generate a block at slj. Specifically, for each
slot a stakeholder Uiis selected as the slot leader with probability piproportional to its stake
registered in the genesis block B0; these assignments are independent between slots. In this static
stake case, the genesis block as well as the procedure for selecting slot leaders are determined by
an ideal functionality FD,F
LS , defined in Figure 1. This functionality is parameterized by the list
{(vk1, s1),...,(vkn, sn)}assigning to each stakeholder its respective stake, a distribution Dthat
provides auxiliary information ρand a leader selection function Fdefined below.
11

Definition 4.7 (Leader Selection Process).Aleader selection process with respect to stakeholder
distribution S={(vk1, s1),...,(vkn, sn)},(D,F)is a pair consisting of a distribution and a de-
terministic function such that, when ρ← D it holds that for al l slj∈ {sl1, . . . , slR},F(S, ρ, slj)
outputs Ui∈ {U1, . . . , Un}with probability
pi=si
Pn
k=1 sk
where siis the stake held by stakeholder Ui(we call this “weighing by stake”); furthermore the
family of random variables {F(S, ρ, slj)}R
j=1 are independent.
We note that sampling proportional to stake can be implemented in a straightforward manner.
For instance, a simple process operates as follows. Let ˜pi=si/Pn
j=isj. For each i= 1, . . . , n −1,
provided that no stakeholder has yet been selected, the process flips a ˜pi-biased coin; if the result
of the coin is 1, the party Uiis selected for the slot and the process is complete. (Note that ˜pn= 1,
so the process is certain to complete with a unique leader.) When we implement this process as
a function F(·), sufficient randomness must be allocated to simulate the biased coin flips. If we
implement the above with λprecision for each individual coin flip, then selecting a stakeholder will
require ndlog λerandom bits in total. Note that using a pseudorandom number generator (PRG)
one may use a shorter “seed” string and then stretch it using the PRG to the appropriate length.
Functionality FD,F
LS [mode]
FD,F
LS [mode] incorporates the diffuse and key/transaction functionality FD+KT from Section 2 and
is parameterized by the public keys and respective stakes of the initial stakeholders S0=
{(U1, s1),...,(Un, sn)}, a distribution Dand a function Fso that (D,F) is a leader selection process.
In addition, FD,F
LS [mode] is parameterized by mode, which determines how signature verification keys
are generated. When FD,F
LS [mode] is instantiated with mode =SIG (resp. mode =FDSIG ) it is denoted
FD,F
LS [SIG] (resp. FD,F
LS [FDSIG]). FD,F
LS interacts with stakeholders as follows:
•Signature Key Pair Generation: FD,F
LS [SIG] generates signing and verification keys ski,vkifor
stakeholder Uiby executing KG(1κ) for i= 1, . . . , n.FD,F
LS [FDSIG] generates (ski,vki) by querying
FDSIG (Figure 3) with (KeyGen, sidi) on behalf of Ui(with a unique session identifier sidirelated
to Ui) and setting (ski=sidi,vki=vi) (received from FDSIG as response) for i= 1, . . . , n.
FD,F
LS [mode] sets S0
0={(vk1, s1),...,(vkn, sn)}.
•Genesis Block Generation Upon receiving (genblock req, Ui) from stakeholder Ui,FD,F
LS pro-
ceeds as follows. If ρhas not been set, FD,F
LS samples ρ← D. In any case, FD,F
LS sends
(genblock,S0
0, ρ, F) to Ui.
•Signatures and Verification.FD,F
LS [FDSIG] provides access to the FDSIG interface.
Figure 1: Functionality FD,F
LS [mode].
A Protocol in the FD,F
LS [mode]-hybrid model. We start by describing a simple PoS based
blockchain protocol considering static stake in the FD,F
LS [SIG]-hybrid model, i.e., where the genesis
block B0(and consequently the slot leaders) are determined by the ideal functionality FD,F
LS [SIG].
FD,F
LS [SIG] provides the stakeholders with a genesis block containing a stake distribution indexed by
signature verification keys generated by a EUF-CMA signature scheme, while FD,F
LS [FDSIG] obtains
such keys from a signature ideal functionality FDSIG . This subtle difference comes into play when
12

describing an ideal version of πSPoS used in an intermediate hybrid argument of the security proof,
which will be discussed in Section 4.2. The stakeholders U1, . . . , Uninteract among themselves and
with FD,F
LS through Protocol πSPoS described in Figure 2.
The protocol relies on a maxvalidS(C,C) function that chooses a chain given the current chain
Cand a set of valid chains Cthat are available in the network. In the static case we analyze the
simple “longest chain” rule. (In the dynamic case the rule is parameterized by a common chain
length; see Section 5.)
Function maxvalid(C,C): Returns the longest chain from C∪ {C}. Ties are broken in
favor of C, if it has maximum length, or arbitrarily otherwise.
Protocol πSPoS
πSPoS is a protocol run by stakeholders U1, . . . , Uninteracting with FD,F
LS [SIG] over a sequence of slots
S={sl1, . . . , slR}.πSPoS proceeds as follows:
1. Initialization Stakeholder Ui∈ {U1, . . . , Un}, receives from the key registration interface its
public and secret key. Then it receives the current slot from the diffuse interface and in case it
is sl1it sends (genblock req, Ui) to FD,F
LS [SIG], receiving (genblock,S0, ρ, F) as answer. Uisets the
local blockchain C=B0= (S0, ρ) and the initial internal state st =H(B0). Otherwise, it receives
from the key registration interface the initial chain C, sets the local blockchain to Cand the initial
internal state st =H(head(C)).
2. Chain Extension For every slot slj∈S, every stakeholder Uiperforms the following steps:
(a) Collect all valid chains received via broadcast into a set C, verifying that for every chain C0∈
Cand every block B0= (st0, d0, sl0, σ0)∈ C0it holds that Vrfvk0(σ0,(st0, d0, sl0)) = 1, where vk0
is the verification key of the stakeholder U0=F(S0, ρ, sl0). Uicomputes C0=maxvalid(C,C),
sets C0as the new local chain and sets state st =H(head(C0)).
(b) If Uiis the slot leader determined by F(S0, ρ, slj), it generates a new block B= (st, d, slj, σ)
where st is its current state, d∈ {0,1}∗is the transaction data and σ=Signski(st, d, slj)
is a signature on (st, d, slj). Uicomputes C0=C|B, broadcasts C0, sets C0as the new local
chain and sets state st =H(head(C0)).
3. Transaction generation Given a transaction template tx,Uireturns σ=Signski(tx), provided
that tx is consistent with the state of the ledger in the view of Ui.
Figure 2: Protocol πSPoS.
4.2 Security Analysis of an Ideal Protocol
As a first step of the security analysis of πSPoS, we will introduce an idealized protocol πiSPoS and
present an intermediate hybrid argument that shows that it is computationally indistinguishable
from πSPoS. Instead of relying on FD,F
LS [SIG] and an EUF-CMA signature scheme, πiSPoS operates
with an ideal signature scheme. To that end, πiSPoS interacts with FD,F
LS [FDSIG] for obtaining signing
and verification keys for the ideal signature scheme employed in the protocol. In the next sessions,
we will prove that πiSPoS is secure through a series of combinatorial arguments. The reason we
first present this hybrid is that we intend to insulate these combinatorial arguments from the
specific details of the underlying signature schemes used to instantiate πSPoS and the biases that
these schemes might introduce in the distributions of πSPoS, concentrating instead on idealized
executions where signature schemes are perfectly realized, which reflects the true nature of our
protocol.
13

Functionality FDSIG
FDSIG interacts with stakeholders as follows:
•Key Generation Upon receiving a message (KeyGen, sid) from a stakeholder Ui, verify that sid =
(Ui, sid0) for some sid0. If not, then ignore the request. Else, hand (KeyGen, sid) to the adversary.
Upon receiving (VerificationKey, sid, v) from the adversary, output (VerificationKey, sid, v) to Ui,
and record the pair (Ui, v).
•Signature Generation Upon receiving a message (Sign, sid, m) from Ui, verify that sid =
(Ui, sid0) for some sid0. If not, then ignore the request. Else, send (Sign, sid, m) to the ad-
versary. Upon receiving (Signature, sid, m, σ) from the adversary, verify that no entry (m, σ, v, 0)
is recorded. If it is, then output an error message to Uiand halt. Else, output (Signature, sid, m, σ)
to Ui, and record the entry (m, σ, v, 0).
•Signature Verification Upon receiving a message (Verify, sid, m, σ, v0) from some stakeholder Ui,
hand (Verify, sid, m, σ, v0) to the adversary. Upon receiving (Verified, sid, m, φ) from the adversary
do:
1. If v0=vand the entry (m, σ, v, 1) is recorded, then set f= 1. (This condition guarantees
completeness: If the verification key v0is the registered one and σis a legitimately generated
signature for m, then the verification succeeds.)
2. Else, if v0=v, the signer is not corrupted, and no entry (m, σ0, v, 1) for any σ0is recorded,
then set f= 0 and record the entry (m, σ, v, 0). (This condition guarantees unforgeability: If
v0is the registered one, the signer is not corrupted, and never signed m, then the verification
fails.)
3. Else, if there is an entry (m, σ, v0, f 0) recorded, then let f=f0. (This condition guaran-
tees consistency: All verification requests with identical parameters will result in the same
answer.)
4. Else, let f=φand record the entry (m, σ, v0, φ).
Output (Verified, sid, m, f) to Ui.
Figure 3: Functionality FDSIG.
First, in Figure 3, we present Functionality FDSIG as defined in [14], where it is also shown that
EUF-CMA signature schemes realize FDSIG. Notice that this fact will be used to show that our
idealized protocol can actually be realized based on practical digital signature schemes such DSA
and ECDSA) and ultimately that πiSPoS is indistinguishable from πSPoS.
The idealized protocol πiSPoS is run by the stakeholders interacting with FD,F
LS [FDSIG] and FDSIG .
Basically, πiSPoS behaves exactly as πSPoS except for calls to Vrfvk(σ) and Signsk (m). Namely, instead
of locally computing Signski(m), Uisends (Sign, sid, m) to FDSIG , receiving (Signature, sid, m, σ)
and outputting σas the signature. Moreover, instead locally computing Vrfvk0(σ, m), Uisends
(Verify, sidi, m, σ, v0) to FDSIG (where v0corresponds to verification key vk0), outputting the value
freceived in message (Verified, sidi, m, f). Protocol πiSPoS is described in Figure 4. This idealized
description will be further developed when arguing about the dynamic stake case, where additional
building blocks must be considered in the idealized protocol.
The following proposition is an immediate corollary of the results in [14] showing that EUF-CMA
signature schemes realize FDSIG.
Proposition 4.8. For each PPT A,Zit holds that there is a PPT Sso that EXECP,FD,F
LS [SIG]
πSPoS,A,Z(λ)
and EXECP,FD,F
LS [FDSIG]
πiSPoS,S,Z(λ)are computational ly indistinguishable.
In light of the above proposition in the remaining of the analysis we will focus on the properties
14

Protocol πiSPoS
πiSPoS is a protocol run by stakeholders U1, . . . , Uninteracting with FD,F
LS [FDSIG] over a sequence of slots
S={sl1, . . . , slR}.πiSPoS proceeds as follows:
1. Initialization Stakeholder Ui∈ {U1, . . . , Un}, receives from the key registration interface its
public and secret key. Then it receives the current slot from the diffuse interface and in case it is
sl1it sends (genblock req, Ui) to FD,F
LS [FDSIG], receiving (genblock,S0, ρ, F) as answer. Uisets the
local blockchain C=B0= (S0, ρ) and the initial internal state st =H(B0). Otherwise, it receives
from the key registration interface the initial chain C, sets the local blockchain to Cand the initial
internal state st =H(head(C)).
2. Chain Extension For every slot slj∈S, every stakeholder Uiperforms the following steps:
(a) Collect all valid chains received via broadcast into a set C, verifying that for every
chain C0∈Cand every block B0= (st0, d0, sl0, σ0)∈ C0it holds that FDSIG answers
with (Verified, sid, (st0, d0, sl0),1) upon being queried with (Verify, sid, (st0, d0, sl0), σ0,vk0),
where vk0is the verification key of the stakeholder U0=F(S0, ρ, sl0). Uicomputes
C0=maxvalid(C,C), sets C0as the new local chain and sets state st =H(head(C0)).
(b) If Uiis the slot leader determined by F(S0, ρ, slj), it generates a new block B= (st, d, slj, σ)
where st is its current state, d∈ {0,1}∗is the transaction data and σis obtained from
FDSIG’s answer (Signature, sid, (st, d, slj), σ) upon being queried with (Sign, sidi,(st, d, slj)).
Uicomputes C0=C|B, broadcasts C0, sets C0as the new local chain and sets state st =
H(head(C0)).
3. Transaction generation Given a transaction template tx,Uireturns σobtained from FDSIG’s
answer (Signature, sidi, tx, σ) upon being queried with (Sign, sidi, tx), provided that tx is consistent
with the state of the ledger in the view of Ui.
Figure 4: Protocol πiSPoS.
of the protocol πiSPoS (note that this implication does not apply to any5possible property one
might consider in an execution for πiSPoS; nevertheless the properties we will prove for πiSPoS are all
verifiable by the environment Zand as a result they can be inherited by πSPoS due to proposition
).
4.3 Forkable Strings
In our security arguments we routinely use elements of {0,1}nto indicate which slots—among
a particular window of slots of length n—have been assigned to adversarial stakeholders. When
strings have this interpretation we refer to them as characteristic strings.
Definition 4.9 (Characteristic String).Fix an execution with genesis block B0, adversary A, and
environment Z. Let S={sli+1 , . . . , sli+n}denote a sequence of slots of length |S|=n. The
characteristic string w∈ {0,1}nof Sis defined so that wk= 1 if and only if the adversary controls
the slot leader of slot sli+k. For such a characteristic string w∈ {0,1}∗we say that the index iis
adversarial if wi= 1 and honest otherwise.
We start with some intuition on our approach to analyze the protocol. Let w∈ {0,1}nbe a
characteristic string for a sequence of slots S. Consider two observers that (i.) go offline immediately
prior to the commencement of S, (ii.) have the same view C0of the current chain prior to the
commencement of S, and (iii.) come back online at the last slot of Sand request an update of their
chain. A fundamental concern in our analysis is the possibility that such observers can be presented
5An example of such a property would be a property testing a non-trivial fact about the parties’ private states.
15

with a “diverging” view over the sequence S: specifically, the possibility that the adversary can
force the two observers to adopt two different chains C1,C2whose common prefix is C0.
We observe that not all characteristic strings permit this. For instance the (entirely honest)
string 0nensures that the two observers will adopt the same chain Cwhich will consist of nnew
blocks on top of the common prefix C0. On the other hand, other strings do not guarantee such
common extension of C0; in the case of 1n, it is possible for the adversary to produce two completely
different histories during the sequence of slots Sand thus furnish to the two observers two distinct
chains C1,C2that only share the common prefix C0. In the remainder of this section, we establish
that strings that permit such “forkings” are quite rare—indeed, we show that they have density
2−Ω(√n)so long as the fraction of adversarial slots is 1/2−.
To reason about such “forkings” of a characteristic string w∈ {0,1}n, we define below a formal
notion of “fork” that captures the relationship between the chains broadcast by honest slot leaders
during an execution of the protocol πiSPoS. In preparation for the definition, we recall that honest
players always choose to extend a maximum length chain among those available to the player on
the network. Furthermore, if such a maximal chain Cincludes a block Bpreviously broadcast by
an honest player, the prefix of Cprior to Bmust entirely agree with the chain (terminating at B)
broadcast by this previous honest player. This “confluence” property follows immediately from the
fact that the state of any honest block effectively commits to a unique chain beginning at the genesis
block. To conclude, any chain Cbroadcast by an honest player must begin with a chain produced
by a previously honest player (or, alternatively, the genesis block), continue with a possibly empty
sequence of adversarial blocks and, finally, terminate with an honest block. It follows that the
chains broadcast by honest players form a natural directed tree. The fact that honest players
reliably broadcast their chains and always build on the longest available chain introduces a second
important property of this tree: the “depths” of the various honest blocks added by honest players
during the protocol must all be distinct.
Of course, the actual chains induced by an execution of πiSPoS are comprised of blocks containing
a variety of data that are immaterial for reasoning about forking. For this reason the formal notion
of fork below merely reflects the directed tree formed by the relevant chains and the identities of
the players—expressed as indices in the string w—responsible for generating the blocks in these
chains.
Forks and forkable strings. We define, below, the basic combinatorial structures we use to
reason about the possible views observed by honest players during a protocol execution with this
characteristic string.
Definition 4.10 (Fork).Let w∈ {0,1}nand let H={i|wi= 0}denote the set of honest indices.
Afork for the string wis a directed, rooted tree F= (V, E)with a labeling `:V→ {0,1, . . . , n}so
that
•each edge of Fis directed away from the root;
•the root r∈Vis given the label `(r)=0;
•the labels along any directed path in the tree are strictly increasing;
•each honest index i∈His the label of exactly one vertex of F;
•the function d:H→ {1, . . . , n}, defined so that d(i)is the depth in Fof the unique vertex v
for which `(v) = i, is strictly increasing. (Specifically, if i, j ∈Hand i<j, then d(i)<d(j).)
16

w=0
1
1
2
2
0
3
1
4
4
0
5
0
6
1 1
8
0
90
ˆ
t
t
Figure 5: A fork Ffor the string w= 010100110; vertices appear with their labels and honest
vertices are highlighted with double borders. Note that the depths of the (honest) vertices associated
with the honest indices of ware strictly increasing. Two tines are distinguished in the figure: one,
labeled ˆ
t, terminates at the vertex labeled 9 and is the longest tine in the fork; a second tine t
terminates at the vertex labeled 3. The quantity gap(t) indicates the difference in length between t
and ˆ
t; in this case gap(t) = 4. The quantity reserve(t) = |{i|`(v)< i ≤ |w|and wi= 1}| indicates
the number of adversarial indices appearing after the label of the last honest vertex vof the tine;
in this case reserve(t) = 3. As each leaf of Fis honest, Fis closed.
As a matter of notation, we write F`wto indicate that Fis a fork for the string w. We say that
a fork is trivial if it contains a single vertex, the root.
Definition 4.11 (Tines, depth, and height; the ∼relation).A path in a fork Foriginating at the
root is called a tine. For a tine twe let length(t)denote its length, equal to the number of edges
on the path. For a vertex v, we let depth(v)denote the length of the (unique) tine terminating at
v. The height of a fork (as usual for a tree) is defined to be the length of the longest tine.
We overload the notation `() so that it applies to tines, by defining `(t),`(v), where vis the
terminal vertex on the tine t. For two tines t1and t2of a fork F, we write t1∼t2if they share an
edge. Note that ∼is an equivalence relation on the set of nontrivial tines; on the other hand, if t
denotes the “empty” tine consisting solely of the root vertex then t6∼ tfor any tine t.
If a vertex vof a fork is labeled with an adversarial index (i.e., w`(v)= 1) we say that the vertex
is adversarial; otherwise, we say that the vertex is honest. For convenience, we declare the root
vertex to be honest. We extend this terminology to tines: a tine is honest if it terminates with an
honest vertex and adversarial otherwise. By this convention the empty tine tis honest.
See Figure 5 for an example, which also demonstrates some of the quantities defined above and
in the remainder of this section. The fork shown in the figure reflects an execution in which (i.)
the honest player associated with the first slot builds directly on the genesis block (as it must),
(ii.) the honest player associated with the third slot is shown a chain of length 1 produced by the
adversarial player of slot 2 (in addition to the honestly generated chain of step (i.)), which it elects
to extend, (iii.) the honest player associated with slot 5 is shown a chain of length 2 building on
the chain of step (i.) augmented with a further adversarial block produced by the player of slot 4,
etc.
Definition 4.12. We say that a fork is flat if it has two tines t16∼ t2of length equal to the height
of the fork. A string w∈ {0,1}∗is said to be forkable if there is a flat fork F`w.
Note that in order for an execution of πiSPoS to yield two entirely disjoint chains of maximum
length, the characteristic string associated with the execution must be forkable. Our goal is to
establish the following upper bound on the number of forkable strings.
17

Theorem 4.13. Let ∈(0,1) and let wbe a string drawn from {0,1}nby independently assigning
each wi= 1 with probability (1 −)/2. Then Pr[wis forkable] = 2−Ω(√n).
Note that in subsequent work, Russell et al. [37] improved this bound to 2−Ω(n).
Structural features of forks: closed forks, prefixes, reach, and margin. We begin by
defining a natural notion of inclusion for two forks:
Definition 4.14 (Fork prefixes).If wis a prefix of the string w0∈ {0,1}∗,F`w, and F0`w0,
we say that Fis a prefix of F0, written FvF0, if Fis a consistently-labeled subgraph of F0.
Specifically, every vertex and edge of Fappears in F0and, furthermore, the labels given to any
vertex appearing in both Fand F0are identical.
If FvF0, each tine of Fappears as the prefix of a tine in F0. In particular, the labels appearing
on any tine terminating at a common vertex are identical and, moreover, the depth of any honest
vertex appearing in both Fand F0is identical.
In many cases, it is convenient to work with forks that do not “commit” anything beyond final
honest indices.
Definition 4.15 (Closed forks).A fork is closed if each leaf is honest. By convention the trivial
fork, consisting solely of a root vertex, is closed.
Note that a closed fork has a unique longest tine (as all maximal tines terminate with an honest
vertex, and these must have distinct depths). Note, additionally, that if ˇwis a prefix of wand
F`w, then there is a unique closed fork ˇ
F`ˇwfor which ˇ
FvF. In particular, taking ˇw=w, we
note that for any fork F`w, there is a unique closed fork F`wfor which FvF; in this case we
say that Fis the closure of F.
Definition 4.16 (Gap, reserve and reach).Let F`wbe a closed fork and let ˆ
tdenote the (unique)
tine of maximum length in F. We define the gap of a tine t, denoted gap(t), to be the difference in
length between ˆ
tand t; thus
gap(t) = length(ˆ
t)−length(t).
We define the reserve of a tine tto be the number of adversarial indices appearing in wafter the
last index in t; specifically, if tis given by the path (r, v1, . . . , vk), where ris the root of F, we define
reserve(t) = |{i|wi= 1 and i>`(vk)}|.
We remark that this quantity depends both on Fand the specific string wassociated with F. Finally,
for a tine twe define
reach(t) = reserve(t)−gap(t).
Definition 4.17 (Margin).For a closed fork F`wwe define ρ(F)to be the maximum reach taken
over all tines in F:
ρ(F) = max
treach(t).
Likewise, we define the margin of F, denoted µ(F), to be the “penultimate” reach taken over edge-
disjoint tines of F: specifically,
margin(F) = µ(F) = max
t16∼t2min{reach(t1),reach(t2)}.(1)
18

We remark that the maxima above can always obtained by honest tines. Specifically, if tis an
adversarial tine of a fork F`w, reach(t)≤reach(t), where tis the longest honest prefix of t.
As ∼is an equivalence relation on the nonempty tines, it follows that there is always a pair of
(edge-disjoint) tines t1and t2achieving the maximum in the defining equation (1) which satisfy
reach(t1) = ρ(F)≥reach(t2) = µ(F).
The relevance of margin to the notion of forkability is reflected in the following proposition.
Proposition 4.18. A string wis forkable if and only if there is a closed fork F`wfor which
margin(F)≥0.
Proof. If whas no honest indices, then the trivial fork consisting of a single root node is flat, closed,
and has non-negative margin; thus the two conditions are equivalent. Consider a forkable string w
with at least one honest index and let ˆ
idenote the largest honest index of w. Let Fbe a flat fork
for wand let F`wbe the closure of F(obtained from Fby removing any adversarial vertices from
the ends of the tines of F). Note that the tine ˆ
tcontaining ˆ
iis the longest tine in F, as this is the
largest honest index of w. On the other hand, Fis flat, in which case there are two edge-disjoint
tines t1and t2with length at least that of ˆ
t. The prefixes of these two tines in Fmust clearly have
reserve no less than gap (and hence non-negative reach); thus margin(F)≥0 as desired.
On the other hand, suppose whas a closed fork with margin(F)≥0, in which case there are
two edge-disjoint tines of F,t1and t2, for which reach(ti)≥0. Then we can produce a flat fork by
simply adding to each tia path of gap(ti) vertices labeled with the subsequent adversarial indices
promised by the definition of reserve().
In light of this proposition, for a string wwe focus our attention on the quantities
ρ(w) = max
F`w,
Fclosed
ρ(F), µ(w) = max
F`w,
Fclosed
µ(F),
and, for convenience,
m(w)=(ρ(w), µ(w)) .
Note that this overloads the notation ρ(·) and µ(·) so that they apply to both forks and strings,
but the setting will be clear from context. We remark that the definitions do not guarantee a priori
that ρ(w) and µ(w) can be achieved by the same fork, though this will be established in the lemma
below. In any case, it is clear that ρ(w)≥0 and ρ(w)≥µ(w) for all strings w; furthermore, by
Proposition 4.18 a string wis forkable if and only if µ(w)≥0. We refer to µ(w) as the margin of
the string w.
In preparation for the proof of Theorem 4.13, we establish a recursive description for these
quantities.
Lemma 4.19. m() = (0,0) and, for al l nonempty strings w∈ {0,1}∗,
m(w1) = (ρ(w)+1, µ(w) + 1) ,and
m(w0) = 






(ρ(w)−1,0) if ρ(w)> µ(w)=0,
(0, µ(w)−1) if ρ(w)=0,
(ρ(w)−1, µ(w)−1) otherwise.
Furthermore, for every string w, there is a closed fork Fw`wfor which m(w)=(ρ(Fw), µ(Fw)).
19

Proof. The proof proceeds by induction. If w=, define Fto be the trivial fork; F`wis the
unique closed fork for this string and m() = (0,0) = (ρ(F), µ(F)), as desired.
In general, we consider m(w0) for a string w0=wx—where w∈ {0,1}∗and x∈ {0,1}; the
argument recursively expands m(w0) in terms of m(w) and the value of the last symbol x. In each
case, we consider the relationship between two closed forks FvF0where F`wand F0`w0=wx.
In the case where x= 1, we must have F=F0as graphs, because the forks are assumed to
be closed; it is easy to see that the reach of any tine tof F`whas increased by exactly one
when viewed as a tine of F0`w0. We write reachF0(t) = reachF(t) + 1, where we introduce the
notation reach() to denote the reach in a particular fork. It follows that ρ(F0) = ρ(F) + 1 and
µ(F0) = µ(F) + 1. If F∗`w0is a closed fork for which ρ(F∗) = ρ(w0), note that F∗may be
treated as a fork for wand, applying the argument above, we find that ρ(w0)≤ρ(w) + 1. A similar
argument implies that µ(w0)≤µ(w) + 1. On the other hand, by induction there is a fork Fwfor
which m(w)=(ρ(Fw), µ(Fw)) and hence m(w0)≥(ρ(w)+1, µ(w) + 1). We conclude that
m(w0)=(ρ(w)+1, µ(w) + 1) .(2)
Moreover, m(w0)=(ρ(Fw), µ(Fw)), where Fwis treated as a fork for w0=w1.
The case when x= 0 is more delicate. As above, we consider the relationship between two
closed forks F`wand F0`w0=w0 for which FvF0. Here F0is necessarily obtained from F
by appending a path labeled with a string of the form 1a0 to the end of a tine tof F. (In fact,
it is easy to see that we may always assume that this is appended to an honest tine.) In order
for this to be possible, gap(t)≤reserve(t) (which is to say that reach(t)≥0) and, in particular,
gap(t)≤a≤reserve(t): for the first inequality, note that the depth of the new honest vertex
must exceed that of the deepest (honest) vertex in Fand hence a≥gap(t); as for the second
inequality, there are only reserve(t) possible adversarial indices that may be added to tand hence
a≤reserve(t). We define the quantity ˜a≥0 by the equation a= gap(t) + ˜aand let t0denote the
tine (of F0) resulting by extending tin this way. We say that ˜ais the parameter for this pair of
forks FvF0.
Of course, every honest tine tof Fis an honest tine of F0and it is clear that reachF0(t) =
reachF(t)−(˜a+ 1), as the length of the longest tine t0in F0exceeds the length of the longest tine
of Fby exactly ˜a+ 1. Note that the reach of the new honest tine t0(in F0) is always 0, as both
gap(t0) and reserve(t0) are zero. It remains to describe how µ(w) and ρ(w) are determined by this
process.
The case ρ(w)> µ(w)=0.By induction, there is a fork Fwfor which m(w) = (ρ(Fw), µ(Fw)).
Let t1and t2be edge-disjoint tines of Fwfor which ρ(Fw) = reach(t1) and µ(Fw) = reach(t2).
Define F0`w0to be the fork obtained by extending the tine t2of Fwwith parameter ˜a= 0
to yield a new tine t0
2in F0. Then reachF0(t1) = ρ(w)−1 and reachF0(t0
2) = 0. It follows that
ρ(w0) ≥ρ(w)−1 and µ(w0) ≥0. We will show that ρ(w0) ≤ρ(w)−1 and that µ(w0) ≤0,
in which case we can conclude that
ρ(w0) = ρ(w)−1 and µ(w0) = 0 .
Moreover, the fork Fw0=F0achieves these statistics, as desired.
We return to establish that ρ(w0) ≤ρ(w)−1 and that µ(w0) ≤0. Let F∗`w0 be a closed
fork for which ρ(w0) = ρ(F∗) and let F`wbe the unique closed fork for which FvF∗;
as above, let ˜adenote the parameter for this extension. Let t∗be an honest tine of F∗so
that reachF∗(t∗) = ρ(w0). If t∗is a tine of F, reachF∗(t∗) = reachF(t∗)−(˜a+ 1) ≤ρ(w)−1.
Otherwise t∗was obtained by extension and reachF∗(t∗)=0≤ρ(w)−1 by assumption. In
20

either case ρ(w0) ≤ρ(w)−1, as desired. It remains to show that µ(w0) ≤0. Now consider
F∗`w0 to be a closed fork for which µ(F∗) = µ(w0). Let t∗
1and t∗
2be two edge-disjoint
honest tines of F∗so that reachF∗(t∗
1) = ρ(F∗) and reachF∗(t∗
2) = µ(F∗) = µ(w0). Let F`w
be the unique closed fork for which FvF∗and let ˜abe the parameter for this extension.
If both t∗
1and t∗
2are tines of F, reachF∗(t∗
i) = reachF(t∗
i)−(˜a+ 1) and, in particular,
reachF(t∗
1)≥reachF(t∗
2). It follows that reachF(t∗
2)≤µ(F)≤µ(w) = 0 and hence that
µ(w0) <0. Otherwise, one of the two tines was the result of extension and has zero reach()
in F∗. As reachF∗(t∗
1)≥reachF∗(t∗
2), in either case it follows that µ(F∗) = reachF∗(t∗
2)≤0,
as desired.
The case ρ(w)=0.By induction, there is a fork Fwfor which m(w)=(ρ(Fw), µ(Fw)). Let t1
and t2be edge-disjoint tines of Fwfor which ρ(Fw) = reach(t1) and µ(Fw) = reach(t2). Define
F0`w0to be the fork obtained by extending the tine t1of Fwwith parameter ˜a= 0 to yield
a new tine t0
1in F0. Then reachF0(t0
1) = 0 and reachF0(t2) = reachF(t2)−1. It follows that
ρ(w0) ≥0 and µ(w0) ≥µ(w)−1. We will show that ρ(w0) ≤0 and that µ(w0) ≤µ(w)−1,
in which case we can conclude that
ρ(w0) = 0 and µ(w0) = µ(w)−1.
Moreover, the fork Fw0=F0achieves these statistics, as desired.
We return to establish that ρ(w0) ≤0 and that µ(w0) ≤µ(w)−1. Let F∗`w0 be a
closed fork for which ρ(w0) = ρ(F∗) and let F`wbe the unique closed fork for which
FvF∗; as above, let ˜adenote the parameter for this extension. Let t∗be an honest tine
of F∗so that reachF∗(t∗) = ρ(w0). Note that t∗cannot be a tine of F; if it were then
reachF∗(t∗) = reachF(t∗)−(˜a+ 1) ≤ρ(w)−1<0 which contradicts ρ(w0) ≥0. Thus t∗
was obtained by extension and reachF∗(t∗) = 0. It remains to show that µ(w0) ≤0. Now let
F∗`w0 be a closed fork for which µ(F∗) = µ(w0). Let t∗
1and t∗
2be two edge-disjoint honest
tines of F∗so that reachF∗(t∗
1) = ρ(F∗) and reachF∗(t∗
2) = µ(F∗) = µ(w0). Let F`wbe the
unique closed fork for which FvF∗and let ˜abe the parameter for this extension. Similarly,
t∗
1cannot be a tine of F; if it were, ρ(F∗) = reachF∗(t∗
1) = reachF(t∗
1)−(˜a+ 1) ≤ρ(F)−1≤
ρ(w)−1<0 which contradicts ρ(F)≥0. It follows that t∗
1must extend a tine t1of Ffor
which reachF(t1) = 0, because extension can only occur for tines of non-negative reach and
ρ(F) = 0 = ρ(w). Thus t∗
2is a tine of Fand t16∼ t∗
2so that reachF(t∗
2)≤µ(F)≤µ(w) and
we conclude that µ(w0) = reachF∗(t∗
2)≤reachF(t∗
2)−1≤µ(w)−1, as desired.
The case ρ(w)>0, µ(w)6= 0.By induction, there is a fork Fwfor which m(w)=(ρ(Fw), µ(Fw)).
Let t1and t2be edge-disjoint tines of Fwfor which ρ(Fw) = reach(t1) and µ(Fw) = reach(t2).
In fact, any extension of Fwwill suffice for the construction; for concreteness, define F0`
w0to be the fork obtained by extending the tine t1of Fwwith parameter ˜a= 0. Then
reachF0(ti) = reachFw(ti)−1. It follows that ρ(w0) ≥ρ(w)−1 and µ(w0) ≥µ(w)−1. We
will show that ρ(w0) ≤ρ(w)−1 and that µ(w0) ≤µ(w)−1, in which case we can conclude
that
ρ(w0) = ρ(w)−1 and µ(w0) = µ(w)−1.
Moreover, the fork Fw0=F0achieves these statistics, as desired.
We return to establish that ρ(w0) ≤ρ(w)−1 and that µ(w0) ≤µ(w)−1. Let F∗`w0 be a
closed fork for which ρ(w0) = ρ(F∗) and let F`wbe the unique closed fork for which FvF∗;
as above, let ˜adenote the parameter for this extension. Let t∗be an honest tine of F∗so that
reachF∗(t∗) = ρ(w0). Note that if t∗is a tine of Fthen reachF∗(t∗) = reachF(t∗)−(˜a+ 1) ≤
21

ρ(w)−1; otherwise t∗is obtained by extension and reachF∗(t∗) = 0 ≤ρ(w)−1, as desired.
(Recall that ρ(w)>0.) It remains to show that µ(w0) ≤µ(w)−1. Now let F∗`w0 be a
closed fork for which µ(F∗) = µ(w0). Let t∗
1and t∗
2be two edge-disjoint honest tines of F∗so
that reachF∗(t∗
1) = ρ(F∗) and reachF∗(t∗
2) = µ(F∗) = µ(w0). Let F`wbe the unique closed
fork for which FvF∗and let ˜abe the parameter for this extension. If both t∗
1and t∗
2are
tines of Fthen reachF∗(t∗
i) = reachF(t∗
i)−(˜a+ 1) and, in particular, reachF(t∗
1)≥reachF(t∗
2)
so that reachF(t∗
2)≤µ(w) and reachF∗(t∗
2)≤µ(w)−1, as desired.
To complete the argument, we consider the case that one of the tines t∗
iarises by extension.
Note that in this case reachF∗(t∗
2)≤0, as either t∗
2is obtained by extension so that it has zero
reach, or t∗
1is obtained by extension so that reachF∗(t∗
2)≤reachF∗(t∗
1) = 0. Here we further
separate the analysis into two cases depending on the sign of µ(w):
•If µ(w)>0, then reachF∗(t∗
2)≤0≤µ(w)−1, as desired.
•If µ(w)<0 then t∗
2cannot be the extension of a tine in F. To see this, suppose to the
contrary that t∗
2extends a tine t2of F; then reachF(t2)≥0. Additionally, t∗
1must be a
tine of F, edge-disjoint from t2, and reachF(t∗
1) = reachF∗(t∗
1) + (˜a+ 1) >0. It follows
that µ(w)≥µ(F)≥0, a contradiction.
The other possibility is that t∗
1is an extension of a tine t1of Fin which case reachF(t1)≥
0. Note that t∗
2is a tine of Fand edge-disjoint from t1; thus min(reachF(t∗
2),reachF(t1)) ≤
µ(F)<0 and reachF(t∗
2)≤µ(F). We conclude that reachF∗(t∗
2) = reachF(t∗
2)−(˜a+1) ≤
µ(w)−1, as desired.
With this recursive description in place, we return to the proof of Theorem 4.13, which we
restate below for convenience.
Theorem 4.13, restated. Let ∈(0,1) and let wbe a string drawn from {0,1}nby independently
assigning each wi= 1 with probability (1 −)/2. Then
Pr[wis forkable]=2−Ω(√n).
Proof of Theorem 4.13. The theorem concerns the probability distribution on {0,1}ngiven by in-
dependently selecting each wi∈ {0,1}so that
Pr[wi= 0] = 1 + 
2= 1 −Pr[wi= 1] .
when wis drawn with this distribution. For the string w1. . . wnchosen with the probability
distribution above, define the random variables
Rt=ρ(w1. . . wt) and Mt=µ(w1. . . wt).
Our goal is to establish that
Pr[wforkable] = Pr[Mn≥0] = 2−Ω(√n).
22

We extract from the statement of Lemma 4.19 some facts about these random variables.
Rt>0 =⇒(Rt+1 =Rt+ 1 if wt+1 = 1,
Rt+1 =Rt−1 if wt+1 = 0; (3)
Mt<0 =⇒(Mt+1 =Mt+ 1 if wt+1 = 1,
Mt+1 =Mt−1 if wt+1 = 0; (4)
Rt=0=⇒






Rt+1 = 1 if wt+1 = 1,
Rt+1 = 0 if wt+1 = 0,
Mt+1 <0 if wt= 0.
(5)
In light of the properties (3) above, the random variables Rtare quite well-behaved when
positive—in particular, considering the distribution placed on each wi, they simply follow the
familiar biased random walk of Figure 6. Likewise, considering the properties (4), the random
variables Mtfollow a biased random walk when negative. The remainder of the proof combines
these probability laws with (5) and the fact that Mt() ≤Rt() to establish that Mn<0 with high
probability.
0
−1
··· 1···
p
qq
pp
q
p
q
Figure 6: The simple biased walk where p= (1 + )/2 and q= 1 −p.
We recall two basic facts about the standard biased walk associated with the Markov chain of
Figure 6. Let Zi∈ {±1}(for i= 1,2, . . .) denote a family of independent random variables for
which Pr[Zi= 1] = (1 −)/2. Then the biased walk given by the variables Yt=Pt
iZihas the
following properties.
Constant escape probability; gambler’s ruin. With constant probability, depending only on
,Yt6= 1 for all t > 0. In general, for each k > 0,
Pr[∃t, Yt=k] = αk,(6)
for a constant α < 1 depending only on . (In fact, the constant αis (1 −)/(1 + ); see, e.g.,
[25, Chapter 12] for a complete development.)
Concentration (the Chernoff bound). Consider Tsteps of the biased walk beginning at state
0; then the resulting value is tightly concentrated around −T . Specifically, E[YT] = −T and
Pr YT>−T
2= 2−Ω(T).(7)
(The constant hidden in the Ω() notation depends only on . See, e.g., [1, Cor. A.1.14].)
Partitioning the string w, we write w=w(1) ···w(√n)where w(t)=w1+at−1. . . watand at=dt√ne,
for t= 0,1, . . .. Let R(0) = 0 and R(t)=Rat; similarly define M(0) = 0 and M(t)=Mat. Fix δ
to be a small constant.
We define three events based on the random variables R(t)and M(t):
23

Hot We let Hottdenote the event that R(t)≥δ√nand M(t)≥ −δ√n.
Volatile We let Voltdenote the event that −δ√n≤M(t)≤R(t)< δ√n.
Cold We let Coldtdenote the event that M(t)<−δ√n.
Note that for each t, exactly one of these events occurs—they partition the probability space. Then
we will establish that
Pr[Coldt+1 |Coldt]≥1−2−Ω(√n),(8)
Pr[Coldt+1 |Volt]≥Ω(),(9)
Pr[Hott+1 |Volt]≤2−Ω(√n).(10)
Cold
Vol
Hot
≈1
θ(1)
θ(1) ≈0
Figure 7: An illustration of the transitions between Cold,Vol, and Hot.
Note that the event Vol0occurs by definition. Assuming these inequalities, we observe that the
system is very likely to eventually become cold, and stay that way. In this case, Cold√noccurs,
M◦
n< δ√n < 0, and wis not forkable. Specifically, note that the probability that the system
ever transitions from volatile to hot is no more than 2−Ω(√n)(as transition from Vol to Hot is
bounded above by 2−Ω(√n), and there are no more than √npossible transition opportunities).
Note, also, that while the system is volatile, it transitions to cold with constant probability during
each period. In particular, the probability that the system is volatile for the entire process is no
more that 2−Ω(√n). Finally, note that the probability that the system ever transitions out of the
cold state is no more than 2−Ω(√n)(again, there are at most √npossible times when this could
happen, and any individual transition occurs with probability 2−Ω(√n)). It follows that the system
is cold at the end of the process with probability 1 −2−Ω(√n).
It remains to establish the three inequalities (8), (9), and (10).
Inequality (8):This follows directly from (3) and (6). Specifically, in light of (4) the random
variables Mifollow the probability law of the simple biased walk when they are negative.
Conditioned on M(t)=Mat<−δ√n, the probability that any future Msever climbs to the
value −1 is no more than α−δ√n= 2−Ω(n), as desired. (Here α < 1 is a fixed constant that
depends only on .)
24

Inequality (9):This follows from (3), (5), (6), and (7). Specifically, conditioned on Volt,R(t)≤
δ√n. Recall from (3) that the random variables Rifollow the probability law of the simple
biased walk when they are positive. Let Dbe the event that Ri>0 for all at≤i<at+2δ√n.
According to (7), then, where we take T= 2δ√n, Pr[D]≤2−Ω(√n). With near certainty, then,
the random variables Rivisit the value 0 during this period. Observe that if Ri= 0 then,
by (5), Mi+1 ≤ −1 with constant probability and (conditioned on this), by (6), with constant
probability the subsequent random variables Mjdo not return to the value 0. Additionally,
in light of (7), the probability that there is a sequence wi. . . wjof length at least 2(δ/)√n
for which 

j
X
k=i(1 if wk= 1,
−1 if wk= 0. 
≥ −δ√n
is no more than (√n)22−Ω(√n). It follows that with constant probability, the walk (of Ri)
hits 0, as described above, and then Miterminates at a value less than −δ√n.
Inequality (10):This follows from (3), (5), (6), and (7). Specifically, conditioned on Volt,R(t)≤
δ√n. Recall from (3) that the random variables Rifollow the probability law of the simple
biased walk when they are positive. Let Dbe the event that Ri>0 for all at≤i<at+2δ√n.
According to (7), then, where we take T= 2δ√n, Pr[D]≤2−Ω(√n). With near certainty,
then, the random variables Rivisit the value 0 during this period. Conditioned on D, in order
for Rat+1 ≥δ√nthere must be a sequence of these random variables 0 = Ri, Ri+1, . . . , Rj=
bδ√ncso that none of these take the value 0 except the first. (Such a sequence arises by
taking ito be the last time the variables Rat, . . . visit 0 and jthe first subsequent time
that the sequence is larger than δ√n.) In light of (6), the probability of such a subsequence
appearing at a particular value for iis no more than α−δ√n. It follows that the probability
that Rat+1 ≥δ√nis less than √nα−δ√n= 2−Ω(√n), as desired.
Exact probabilities of forkability for explicit values of n.In order to gain further insight
regarding the density of forkable strings, we exactly computed the probability that a string wdrawn
from the binomial distribution with parameter p∈ {.40, .41, . . . , .50}is forkable for several different
lengths. These results are presented in Figure 8.
4.3.1 Covert adversaries, covert forks, and covertly forkable strings
The general notion of fork defined in Definition 4.10 above reflects the possibility that adversarial
slot leaders may broadcast multiple blocks for a single slot; such adversaries may simultaneously
extend many different chains. While this provides the adversary significant opportunities to in-
terfere with the protocol, it leaves a suspicious “audit trail”—multiple signed blocks for the same
slot—which conspicuously deviates from the protocol.
This motivates our consideration of a restricted class of covert adversaries, who broadcast no
more than one block per slot. Such an adversary may still deviate from the protocol by extend-
ing short chains, but does not produce such suspicious evidence and hence its strategy is more
“deniable”: it can blame network delays for its actions.6
Such an adversary yields a restricted notion of fork, defined below:
6Contrast this with a more general adversary that attempts to fork by signing two different blocks for the same
slot; such an adversary cannot merely blame the network for such a deviation.
25

0.4 0.42 0.44 0.46 0.48 0.5
0
0.2
0.4
0.6
0.8
Binomial distribution parameter
Probability of forkability
Probability of Forkability
n= 500
n= 1000
n= 1500
n= 2000
Figure 8: Graphs of the probability that a string drawn from the binomial distribution is forkable.
Graphs for string lengths n= 500,1000,1500,2000 are shown with parameters .40, .41, . . . , .49, .50.
Definition 4.20. Let F`wbe a fork for a string w∈ {0,1}∗. We say that Fis covert if the
labeling `:V→ {0,1,...,}is injective. In particular, no adversarial index is labeled by more than
one node.
As in the general case, we define a notion of forkable string for such adversaries.
Definition 4.21. We say that a string wis covertly forkable if there is a flat covert fork F`w.
Covert adversaries and forks have much simpler structure than general adversaries. In particu-
lar, a string is covertly forkable if and only if a majority of its indices are adversarial. This provides
an analogue of Proposition 4.18 for covertly forkable strings.
Proposition 4.22. A string w∈ {0,1}nis covertly forkable if and only if wt(w)≥n/2.
Proof. Let wbe a covertly forkable string and F`wa flat covert fork. As Fis flat, there are
two edge disjoint tines, t1and t2, with length equal to height(F) and it follows that the number
of vertices in Fis at least 2 ·height(F) + 1. In this covert case the labeling function is injective,
and it follows that n≥2·height(F). (Recall that the root vertex is labeled by 0, which is not an
index into w.) On the other hand, the height of Fis at least the number of honest indices of w.
We conclude that the length of wis at least twice the number of honest indices, as desired.
If wt(w)≥n/2, we can produce a flat covert fork F`wby placing all honest indices on a
common tine t1and selecting length(t1) adversarial indices to form an edge-disjoint second tine
t2.
As the structure of covertly forkable strings is so simple, an analogue of Theorem 4.13 for the
density of covertly forkable strings follows directly from standard large deviation bounds.
Theorem 4.23. Let ∈(0,1) and let wbe a string drawn from {0,1}nby independently assigning
each wi= 1 with probability (1 −)/2. Then
Pr[wis covertly forkable]=2−Θ(n).
26

Proof. This follows from standard estimates for the cumulative density function of the binomial
distribution.
Exact probabilities of covert forkability for explicit values of n.For comparison with
the general case, we computed the probability that a string drawn from the binomial distribution
is covertly forkable. These results are presented in Figure 9. (Note that these probabilities are
simply appropriate evaluations of the cumulative density function of the binomial distribution.)
Analogous results for the general case appeared in Figure 8.
0.4 0.42 0.44 0.46 0.48 0.5
0
0.2
0.4
Binomial distribution parameter
Probability of covert forkability
Probability of Covert Forkability
n= 500
n= 1000
n= 1500
n= 2000
Figure 9: Graphs of the probability that a string drawn from the binomial distribution is
covertly forkable. Graphs for string lengths n= 500,1000,1500,2000 are shown with parameters
.40, .41, . . . , .49, .50.
4.4 Common Prefix
Recall that the chains constructed by honest players during an execution of πiSPoS correspond to
tines of a fork, as defined and studied in the previous sections. The random assignment of slots to
stakeholders given by FD,F
LS guarantees that the coordinates of the associated characteristic string w
follow the binomial distribution with probability equal to the adversarial stake. Thus Theorem 4.13
establishes that no execution of the protocol πiSPoS can induce two tines (chains) of maximal length
with no common prefix.
In the context of πiSPoS, however, we wish to establish a much stronger common prefix property:
any pair of chains which could, in principle, be presented by the adversary to an honest party must
have a “recent” common prefix, in the sense that removing a small number of blocks from the
shorter chain results in a prefix of the longer chain.
To formally articulate and prove this property, we introduce some further definitions regarding
tines and forks. We borrow the “truncation operator”, described earlier in the paper for chains:
for a tine twe let tdkdenote the tine obtained by removing the last kedges; if length(t)≤k, we
define tdkto consist solely of the root.
27

Definition 4.24 (Viability).Let F`wbe a fork for a string w∈ {0,1}nand let tbe a tine of F.
We say that tis viable if, for all honest indices h≤`(t), we have
d(h)≤length(t).
(Recall that `(t)is the label of the terminal vertex of t.)
If tis viable, an external (honest) observer witnessing the execution at time `(t)—if provided
the tine talong with all honest tines generated up to time `(t)—could conceivably select tvia the
maxvalid() rule. Observe that any honest tine is viable: by definition, the depth of the terminal
vertex of an honest tine exceeds that of all prior honest vertices.
Definition 4.25 (Divergence).Let Fbe a fork for a string w∈ {0,1}∗. For two viable tines t1
and t2of F, define their divergence to be the quantity
div(t1, t2) = min
i(length(ti)−length(t1∩t2)) ,
where t1∩t2denotes the common prefix of t1and t2. We overload this notation by defining divergence
for Fas the maximum over all pairs of viable tines:
div(F) = max
t1, t2viable
tines of F
div(t1, t2).
Finally, define the divergence of wto be the maximum such divergence over all al l possible forks
for w:
div(w) = max
F`wdiv(F).
Observe that if div(t1, t2)≤kand, say, length(t1)≤length(t2), the tine tdk
1is a prefix of t2.
We first establish that a string with large divergence must have a large forkable substring. We
then apply this in Theorem 4.27 below to conclude that characteristic strings arising from πiSPoS
are unlikely to have large divergence and, hence, possess the common prefix property.
Theorem 4.26. Let w∈ {0,1}∗. Then there is forkable substring ˇwof wwith |ˇw| ≥ div(w).
Proof. Consider a fork F`wand a pair of viable tines (t1, t2) for which
div(t1, t2) = div(w).(11)
For simplicity, we assume the tines have been labeled so that `(t1)< `(t2) and further that
|`(t2)−`(t1)|is minimum among all pairs of tines for which (11) holds. (12)
We begin by identifying the substring ˇw; the remainder of the proof is devoted to constructing
a flat fork for ˇwto establish forkability. Let ydenote the last vertex on the tine t1∩t2, as in the
diagram below, and let α,`(y) = `(t1∩t2).
y
t1
t2
28

Let βdenote the smallest honest index of wfor which β≥`(t2), with the convention that if there
is no such index we define β=n+ 1. Observe that, in any case, `(t1)< `(t2) and hence that
β−1≥`(t1). These indices, αand β, distinguish the substring ˇw=wα+1 . . . wβ−1, which will be
the subject of the remainder of the proof. As the function `(·) is strictly increasing along any tine,
observe that
|ˇw|=β−α−1≥`(t1)−`(y)≥length(t1)−length(t1∩t2)
≥min(length(t1),length(t2)) −length(t1∩t2) = div(w),
so ˇwhas the desired length and it suffices to establish that it is forkable.
We briefly summarize the proof before presenting the details. We begin by establishing several
structural properties of the tines t1and t2that follow from the assumptions (11) and (12) above.
To establish that ˇwis forkable we then extract from Fa flat fork (for ˇw) in two steps: (i.) the fork
Fis subjected to some minor restructuring to ensure that all “long” tines pass through y; (ii.) a
flat fork is constructed by treating the vertex yas the root of a portion of the subtree of Flabeled
with indices of ˇw. At the conclusion of the construction, segments of the two tines t1and t2will
yield the required “long, disjoint” tines satisfying the definition of forkable.
We observe, first of all, that the vertex ycannot be adversarial: otherwise it is easy to construct
an alternative fork ˜
F`wand a pair of tines in ˜
Fthat achieve larger divergence. Specifically,
construct ˜
Ffrom Fby adding a new (adversarial) vertex ˜yto Ffor which `(˜y) = `(y), adding an
edge to ˜yfrom the vertex preceding y, and replacing the edge of t1following ywith one from ˜y;
then the other relevant properties of the fork are maintained, but the divergence of the resulting
tines has increased by one. (See the diagram below.)
y
˜y
t1
t2
A similar argument implies that the fork F0`w1. . . wαobtained by including only those vertices
of Fwith labels less than or equal to α=`(y) has a unique vertex of depth depth(y) (namely, y
itself). In the presence of another vertex ˜y(of F0) with depth depth(y), “redirecting” t1through
˜y(as in the argument above) would likewise result in a fork with larger divergence. Note that `(·)
would indeed be increasing along this new tine (resulting from redirecting t1) because `(˜y)≤`(y)
according to the definition of F0. As αis the last index of the string, this additionally implies that
F0has no vertices of depth exceeding depth(y).
We remark that the minimality assumption (12) implies that any honest index hfor which
h < β has depth no more than min(length(t1),length(t2)): specifically,
h < β =⇒d(h)≤min(length(t1),length(t2)) .(13)
To see this, consider an honest index h < β and a tine thfor which `(th) = h. Recall that t1and
t2are viable; as h < `(t2) it follows immediately that d(h)≤length(t2). Similarly, if h≤`(t1)
then d(h)≤length(t1), so it remains to settle the case when `(t1)< h < `(t2): in particular, in
this regime we wish to likewise guarantee that d(h)≤length(t1). For the sake of contradiction,
assume that length(th) = d(h)>length(t1). Considering the tine th, we separately investigate
two cases depending on whether thshares an edge with t1after the vertex y. If, indeed, thand t1
share an edge after the vertex ythen thand t2do not share such an edge, and we observe that
div(th, t2)≥div(t1, t2) while |`(t2)−h|<|`(t2)−`(t1)|which contradicts (12). If, on the other
29

hand, thshares no edge with t1after y, we similarly observe that div(t1, th)≥div(t1, t2) while
|th−`(t1)|<|`(t2)−`(t1)|, which contradicts (12).
In light of the remarks above, we observe that the fork Fmay be “pinched” at yto yield an
essentially identical fork FByC`wwith the exception that all tines of length exceeding depth(y)
pass through the vertex y. Specifically, the fork FByC`wis defined to be the graph obtained
from Fby changing every edge of Fdirected towards a vertex of depth depth(y) + 1 so that it
originates from y. To see that the resulting tree is a well-defined fork, it suffices to check that `(·)
is still increasing along all tines of FByC. For this purpose, consider the effect of this pinching on
an individual tine tterminating at a particular vertex v—it is replaced with a tine tByCdefined so
that:
•If length(t)≤depth(y), the tine tis unchanged: tByC=t.
•Otherwise, length(t)>depth(y) and thas a vertex zof depth depth(y) + 1; note that
`(z)> `(y) because F0contains no vertices of depth exceeding depth(y). Then tByCis
defined to be the path given by the tine terminating at y, a (new) edge from yto z, and the
suffix of tbeginning at z. (As `(z)> `(y) this has the increasing label property.)
Thus the tree FByCis a legal fork on the same vertex set; note that depths of vertices in Fand
FByCare identical.
By excising the tree rooted at yfrom this pinched fork FByCwe may extract a fork for the
string wα+1 . . . wn. Specifically, consider the induced subgraph FyCof FByCgiven by the vertices
{y}∪{z|depth(z)>depth(y)}. By treating yas a root vertex and suitably defining the labels
`yCof FyCso that `yC(z) = `(z)−`(y), this subgraph has the defining properties of a fork for
wα+1 . . . wn. In particular, considering that αis honest it follows that each honest index h > α has
depth d(h)>length(y) and hence labels a vertex in FyC. For a tine tof FByC, we let tyCdenote
the suffix of this tine beginning at y, which forms a tine in FyC. (If length(t)≤depth(y), we define
tyCto consist solely of the vertex y.) Note that tyC
1and tyC
2share no edges in the fork FyC.
Finally, let ˇ
Fdenote the tree obtained from FyCas the union of all tines tof FyCso that all
labels of tare drawn from ˇw(as it appears as a prefix of wα+1 . . . wn), and
length(t)≤max
h≤| ˇw|
hhonest
d(h).
It is immediate that ˇ
F`ˇw. To conclude the proof, we show that ˇ
Fis flat. For this purpose, we
consider the tines tyC
1and tyC
2. As mentioned above, they share no edges in FyC, and hence the
prefixes ˇ
t1and ˇ
t2(of tyC
1and tyC
2) appearing in ˇ
Fshare no edges. We wish to see that these prefixes
have maximum length in ˇ
F, in which case ˇ
Fis flat, as desired. This is immediate for the tine ˇ
t1
because all labels of tyC
1are drawn from ˇwand, considering (13), its depth is at least that of all
relevant honest vertices. As for ˇ
t2, observe that if `(t2) is not honest then β > `(t2) so that, as with
ˇ
t1, the tine ˇ
t2is labeled by ˇwso that the same argument, relying on (13), ensures that ˇ
t2has length
at least that of all relevant honest vertices. If `(t2) is honest, β=`(t2), and the terminal vertex of
tyC
2does not appear in ˇ
F(as it does not index ˇw). In this case, however, length(tyC
2)>d(h) for
any honest index of ˇw, and it follows that length( ˇ
t2) = length(tyC
2)−1 is at least the depth of any
honest index of ˇw, as desired.
Theorem 4.27. Let k, R ∈Nand ∈(0,1). The probability that the πiSPoS protocol, when executed
with a (1 −)/2fraction of adversarial stake, violates the common prefix property with parameter
kthroughout an epoch of Rslots is no more than exp(−Ω(√k) + ln R); the constant hidden by the
Ω() notation depends only on .
30

sketch. Observe that an execution of πiSPoS violates the common prefix property with parameters
k, R precisely when the fork Finduced by this execution has div(F)≥k. Thus we wish to show
that the probability that div(w)≥kis no more than exp(−Ω(√k) + log R). Let Bad denote the
event that div(w)≥k.
It follows from Theorem 4.26 that if div(w)≥k, there is a forkable substring ˇwof length at
least k.
Thus
Pr[common prefix violation] ≤Pr ∃α, β ∈ {1, . . . , R}so that α+k−1≤βand
wα. . . wβis forkable 
≤X
1≤α≤RX
α+k−1≤β≤R
Pr[wα. . . wβis forkable]
| {z }
(∗)
.
Recall that the characteristic string w∈ {0,1}Rfor such an execution of πiSPoS is determined
by assigning each wi= 1 independently with probability (1 −)/2. According to Theorem 4.13
the probability that a string of length tdrawn from this distribution is forkable is no more than
exp(−c√t) for a positive constant c. Note that for any α≥1,
R
X
t=α+k−1
e−c√t≤Z∞
k−1
e−c√tdt = (2/c2)(1 + c√k−1)e−c√k−1=e−Ω(√k)
and it follows that the sum (∗) above is exp(−Ω(√t)). Thus
Pr[common prefix violation] ≤R·exp(−Ω(√k)) ≤exp(ln R−Ω(√k)) ,
as desired.
4.4.1 Common prefix with covert adversaries
We revisit the notion of common prefix in the setting of covert adversaries. We define the covert
divergence of wto be the maximum divergence over all possible covert forks for w:
cdiv(w) = max
F`w
Fcovert
div(F).
As in the setting with general adversaries, we wish to establish that a string with large covert
divergence must have a large covertly forkable substring. A direct analogue of Theorem 4.27 then
implies that characteristic strings arising from πiSPoS are unlikely to have large covert divergence
and, hence, possess the common prefix property against covert adversaries.
We record an analogue of Theorem 4.26 for covert adversaries.
Theorem 4.28. Let w∈ {0,1}∗. Then there is a covertly forkable substring ˇwof wwith |ˇw| ≥
cdiv(w).
Proof. We are more brief, as portions of the proof have direct analogs in the proof of Theorem 4.26.
Consider a covert fork F`wand a pair of viable tines (t1, t2) of Ffor which div(t1, t2) = cdiv(w);
we assume the tines are identified so that `(t1)< `(t2) and, as in the proof of the general case,
assume that this pair of tines minimizes the quantity |`(t2)−`(t1)|among all pairs with divergence
equal to cdiv(w).
31

Let ydenote the last vertex on the tine t1∩t2. In contrast to the setting with a general
adversary, it is not clear that yis honest and this motivates a slightly different choice for the
beginning of the string ˇw: define αto be the largest honest index of won the tine t1∩t2, with the
convention that α= 0 if there is no such index. As in the proof of Theorem 4.26, define βto be
the smallest honest index of wfor which β≥`(t2), with the convention that β=n+ 1 if there is
no such honest index. Then define ˇw=wα+1 . . . wβ−1; as in the proof of Theorem 4.26 it is easy
to confirm that |ˇw|= (β−1) −α≥`(t1)−`(t1∩t2)≥cdiv(w). The remainder of the proof argues
that ˇwis covertly forkable.
As in the proof of Theorem 4.26, the depth d(h) of any honest index h<βis no more
than min(length(t1),length(t2)): if h≤`(t1) this follows directly from the definition of viabil-
ity. Otherwise, `(t1)<h<`(t2) and we consider the tine thlabeled with h: if length(th)≥
min(length(t1),length(t2)) then the tine th, coupled with either t1or t2, would produce a pair
of tines with divergence no less than div(t1, t2), but for which |`(·)−`(·)|is strictly less than
|`(t1)−`(t2)|.
To complete the proof, we define an injective function i:H→A, where Hdenotes the set of
honest indices in {α+1, . . . , β −1}and Athe complement—the set of adversarial indices of ˇw. The
existence of such a function implies that |H| ≤ |A|and hence that ˇwis covertly forkable by the
criterion given in Proposition 4.22. Let A0⊂Adenote the set of adversarial indices of ˇwappearing
as a label on either of the two tines t1and t2. The function iis defined as follows: i(h), for an
honest index h∈H, is the smallest (adversarial) index of A0which labels a vertex at depth equal
to d(h). Assuming that this function is well-defined it is clearly injective, as labels cannot appear
on multiple vertices of a covert fork and depths of honest vertices are pairwise distinct.
To confirm that i(h) is well-defined, note that for any h∈Hwe must have d(α)<d(h)≤
min(length(t1),length(t2)) and hence there is at least one vertex von each of t1and t2with depth
equal to d(h); furthermore, by the defining properties of αand β, this vertex is labeled with
an index of ˇw. If d(h)≤length(t1∩t2), there is a common vertex von these tines for which
length(v) = d(h); note that this vertex cannot be honest by the definition of α, so i(h) = `(v) is
well-defined in this case. If d(h)>length(t1∩t2), the two tines have distinct vertices at depth
d(h), and one of these must then be adversarial—thus i(h) is well-defined in this case as well.
Finally, we remark that the proof of Theorem 4.27 applies with minor adaptations to the covert
case.
Theorem 4.29. Let k, R ∈Nand ∈(0,1). The probability that the πiSPoS protocol, when executed
with a (1 −)/2fraction of adversarial stake and a covert adversary, violates the common prefix
property with parameter kthroughout a period of Ris no more than exp(−Ω(k)+ ln R); the constant
hidden by the Ω() notation depends only on .
Proof. The proof of Theorem 4.27 applies directly; in this case the asymptotics rely on Theorem 4.23
and the following bound applied in a way that the constant cdepends only on .
∞
X
t=k
e−ct ≤Z∞
k−1
e−ct dt =e−Θ(k).
4.5 Chain Growth and Chain Quality
Anticipating these two proofs, we record an additive Chernoff–Hoeffding bound. (See, e.g., [29] for
a proof.)
32

Theorem 4.30 (Chernoff–Hoeffding bound).Let X1, . . . , XTbe independent random variables with
E[Xi] = piand Xi∈[0,1]. Let X=PT
i=1 Xiand µ=PT
i=1 pi=E[X]. Then, for all δ≥0,
Pr[X≥(1 + δ)µ]≤e−δ2
2+δµand Pr[X≤(1 −δ)µ]≤e−δ2
2+δµ.
We will start with the chain growth property.
Theorem 4.31. The πiSPoS protocol satisfies the chain growth property with parameters τ= 1 −
α, s ∈Nthroughout an epoch of Rslots with probability at least 1−exp(−Ω(2s) + ln R)against an
adversary holding an α−portion of the total stake.
Proof. Define Hama(α) to be the event that the Hamming weight ratio of the characteristic string
that corresponds to the slots [a, a +s−1] is no more than α. Given that the adversarial stake
is α−, each of the kslots has probability α−being assigned to the adversary and thus the
probability that the Hamming weight is more than αs drops exponentially in s. Specifically, using
the additive version of the Chernoff bound, we have that Pr[¬Hama(α)] ≤exp(−22s). It follows
that,
Pr[Hamα]≥1−exp(−22s).
Given the above we know that when Hamαhappens there will be at least (1 −α)shonest slots in
the period of srounds. Given that each honest slot enables an honest party to produce a block,
all honest parties will advance by at least that many blocks. Using a union bound, it follows
that the speed coefficient can be set to τ= (1 −α) and it is satisfied with probability at least
1−exp(−22s+ ln(R)).
Having established chain growth we now turn our attention to chain quality. Recall that the
chain quality property with parameters µand `asserts that among every `consecutive blocks in a
chain (possessed by an honest user), the fraction of adversarial blocks is no more than µ.
Theorem 4.32. Let α−be the adversarial stake ratio. The πiSPoS protocol satisfies the chain
quality property with parameters µ(α−) = α/(1 −α)and `∈Nthroughout an epoch of Rslots
with probability at least
1−exp−Ω(2α`) + ln R.
Proof. First, from the proof for chain growth (Theorem 4.31), we know that with high probability a
segment of `rounds will involve at least (1−α)`slots with honest leaders; hence the resulting chain
must advance by at least (1−α)`blocks. By similar reasoning, the adversarial parties are associated
with no more than α` slots, and thus can contribute no more than α` blocks to any particular chain
over this period. It follows that the associated chain possessed by any honest party contains a
fraction α/(1 −α) of adversarial blocks with probability 1−exp(−Ω(2min(α, 1−α)`) + ln R).
5 Our Protocol: Dynamic Stake
5.1 Using a Trusted Beacon
In the static version of the protocol in the previous section, we assumed that stake was static during
the whole execution (i.e., one epoch), meaning that stake changing hands inside a given epoch does
not affect leader election. Now we put forth a modification of protocol πSPoS that can be executed
over multiple epochs in such a way that each epoch’s leader election process is parameterized by
the stake distribution at a certain designated point of the previous epoch, allowing for change in
33

the stake distribution across epochs to affect the leader election process. As before, we construct
the protocol in a hybrid model, enhancing the FD,F
LS ideal functionality to now provide randomness
and auxiliary information for the leader election process throughout the epochs (the enhanced
functionality will be called FD,F
DLS). We then discuss how to implement FD,F
DLS using only FD,F
LS and
in this way reduce the assumption back to the simple common random string selected at setup.
Before describing the protocol for the case of dynamic stake, we need to explain the modifi-
cation of FD,F
LS so that multiple epochs are considered. The resulting functionality, FD,F
DLS, allows
stakeholders to query it for the leader selection data specific to each epoch. FD,F
DLS is parameterized
by the initial stake of each stakeholder before the first epoch e1starts; in subsequent epochs, par-
ties will take into consideration the stake distribution in the latest block of the previous epoch’s
first R−2kslots. Given that there is no predetermined view of the stakeholder distribution, the
functionality FD,F
DLS will provide only a random string and will leave the interpretation according to
the stakeholder distribution to the party that is calling it. The effective stakeholder distribution
is the sequence S1,S2, . . . defined as follows: S1is the initial stakeholder distribution; for slots
{(j−1)R+ 1, . . . , jR}for j≥2 the effective stakeholder Sjis determined by the stake allocation
that is found in the latest block with time stamp at most (j−1)R−2k, provided all honest parties
agree on it, or is undefined if the honest parties disagree on it. The functionality FD,F
DLS is defined
in Figure 10.
Functionality FD,F
DLS
FD,F
DLS incorporates the diffuse and key/transaction functionality from Section 2 and is parameter-
ized by the public keys and respective stakes of the initial (before epoch e1starts) stakeholders
S0={(vk1, s0
1),...,(vkn, s0
n)}a distribution Dand a leader selection function F. In addition, FD,F
DLS
operates as follows:
•Genesis Block Generation Upon receiving (genblock req, Ui) from stakeholder Uiit operates
as functionality FD,F
LS [SIG] on that message.
•Signature Key Pair Generation It operates as functionality FD,F
LS [SIG].
•Epoch Randomness Update Upon receiving (epochrnd req, Ui, ej) from stakeholder Ui, if j≥2
is the current epoch, FD,F
DLS proceeds as follows. If ρjhas not been set, FD,F
DLS samples ρj← D.
Then, FD,F
DLS sends (epochrnd, ρj) to Ui.
Figure 10: Functionality FD,F
DLS.
We now describe protocol πDPoS, which is a modified version of πSPoS that updates its genesis
block B0(and thus the leader selection process) for every new epoch. The protocol also adopts
an adaptation of the static maxvalidSfunction, defined so that it narrows selection to those chains
which share common prefix. Specifically, it adopts the following rule, parameterized by a prefix
length k:
Function maxvalid(C,C). Returns the longest chain from C∪ {C} that does not fork
from Cmore than kblocks. If multiple exist it returns C, if this is one of them, or it
returns the one that is listed first in C.
Protocol πDPoS is described in Figure 11 and functions in the FD,F
DLS-hybrid model.
Remark 1. The modification to maxvalid(·)to not diverge more than kblocks from the last chain
possessed will require stakeholders to be online at least every kslots. The relevance of the rule
34

Protocol πDPoS
πDPoS is a protocol run by a set of stakeholders, initially equal to U1, . . . , Un, interacting with FD,F
DLS over
a sequence of Lslots S={sl1, . . . , slL}.πDPoS proceeds as follows:
1. Initialization Stakeholder Ui∈ {U1, . . . , Un}, receives from the key registration interface its
public and secret key. Then it receives the current slot from the diffuse interface and in case it
is sl1it sends (genblock req, Ui) to FD,F
LS , receiving (genblock,S0, ρ, F) as the answer. Uisets the
local blockchain C=B0= (S0, ρ) and the initial internal state st =H(B0). Otherwise, it receives
from the key registration interface the initial chain C, sets the local blockchain as Cand the initial
internal state st =H(head(C)).
2. Chain Extension For every slot sl ∈S, every online stakeholder Uiperforms the following steps:
(a) If a new epoch ej, with j≥2, has started, Uidefines Sjto be the stakeholder distribution
drawn from the most recent block with time stamp less than jR −2kas reflected in Cand
sends (epochrnd req, Ui, ej) to FD,F
LS , receiving (epochrnd, ρj) as answer.
(b) Collect all valid chains received via broadcast into a set C, verifying that for every chain
C0∈Cand every block B0= (st0, d0, sl0, σ0)∈ C0it holds that Vrfvk0(σ0,(st0, d0, sl0)) = 1,
where vk0is the verification key of the stakeholder U0=F(Sj0, ρj0, sl0) with ej0being the
epoch in which the slot B0belongs (as determined by sl0). Uicomputes C0=maxvalid(C,C),
sets C0as the new local chain and sets state st =H(head(C0)).
(c) If Uiis the slot leader determined by F(Sj, ρj, sl) in the current epoch ej, it generates
a new block B= (st, d, sl, σ) where st is its current state, d∈ {0,1}∗is the data and
σ=Signski(st, d, sl) is a signature on (st, d, sl). Uicomputes C0=C|B, broadcasts C0, sets
C0as the new local chain and sets state st =H(head(C0)).
3. Transaction generation as in protocol πSPoS.
Figure 11: Protocol πDPoS
comes from the fact that as stake shifts over time, it will be feasible for the adversary to corrupt
stakeholders that used to possess a stake majority at some point without triggering Bad1/2and thus
any adversarial chains produced due to such an event should be rejected. It is worth noting that this
restriction can be easily lifted if one can trust honest stakeholders to securely erase their memory;
in such case, a forward secure signature can be employed to thwart any past corruption attempt that
tries to circumvent Bad1/2.
5.2 Simulating a Trusted Beacon
While protocol πDPoS handles multiple epochs and takes into consideration changes in the stake
distribution, it still relies on FD,F
DLS to perform the leader selection process. In this section, we
show how to implement FD,F
DLS through Protocol πDLS, which allows the stakeholders to compute
the randomness and auxiliary information necessary in the leader election.
Recall, that the only essential difference between FD,F
LS and FD,F
DLS is the continuous generation
of random strings ρ2, ρ3, . . . for epochs e2, e3, . . .. The idea is simple, protocol πDLS will use a coin
tossing protocol to generate unbiased randomness that can be used to define the values ρj, j ≥2
bootstrapping on the initial random string and initial honest stakeholder distribution. However,
notice that the adversary could cause a simple coin tossing protocol to fail by aborting. Thus, we
build a coin tossing scheme with “guaranteed output delivery.”
Protocol πDLS is described in Figure 13 and uses a publicly verifiable secret sharing (PVSS)
[39].
As in the static stake case, we need to define an idealized protocol that behaves as if the
35

computationally secure primitives that are employed in the real protocol behave perfectly. Once
again we will base our combinatorial arguments on this idealized version. We remark that we
depart πiSPoS as previously defined, adding further considerations about an ideal execution of the
coin tossing procedure that generates randomness for the leader selection process. The assumption
we will use about the PVSS scheme is that the resulting coin-flipping protocol simulates a perfect
beacon with distinguishing advantage DLS. Simulation here suggests that, in the case of honest
majority, there is a simulator that interacts with the adversary and produces indistinguishable
protocol transcripts when given the beacon value after the commitment stage. We remark that
using [39] as a PVSS, a simulator can achieve simulatability in the random oracle model by taking
advantage of the programmability of the oracle. Using a random oracle is by no means necessary
though and the same benefits may be obtained by a CRS embedded into the genesis block.
Commitments and Coin Tossing. A coin tossing protocol allows two or more parties to obtain
a uniformly random string. A classic approach to construct such a protocol is by using commitment
schemes. In a commitment scheme, a committer carries out a commitment phase, which sends
evidence of a given value to a receiver without revealing it; later on, in an opening phase, the
committer can send that value to the receiver and convince it that the value is identical to the
value committed to in the commitment phase. Such a scheme is called binding if it is hard for
the committer to convince the receiver that he was committed to any value other than the one for
which he sent evidence in the commitment phase, and it is called hiding if it is hard for the receiver
to learn anything about the value before the opening phase. We denote the commitment phase
with randomness rand message mby Com(r, m) and the opening as Open(r, m).
In a standard two-party coin tossing protocol [9], one party starts by sampling a uniformly
random string u1and sending Com(r, u1). Next, the other party sends another uniformly random
string u2in the clear. Finally, the first party opens u1by sending Open(r, u1) and both parties
compute output u=u1⊕u2. Note, however, that in this classical protocol the committer may
selectively choose to “abort” the protocol (by not opening the commitment) once he observes the
value u2. While this is an intrinsic problem of the two-party setting, we can avoid this problem
in the multi-party setting by relying on a verifiable secret sharing scheme and an honest majority
amongst the protocol participants.
Verifiable Secret Sharing (VSS). A secret sharing scheme allows a dealer PDto split a secret
σinto nshares distributed to parties P1, . . . , Pn, such that no adversary corrupting up to tparties
can recover σ. In a Verifiable Secret Sharing (VSS) scheme [22], there is the additional guarantee
that the honest parties can recover σeven if the adversary corrupts the shares held by the parties
that it controls and even if the dealer itself is malicious. We define a VSS scheme as a pair of efficient
dealing and reconstruction algorithms (Deal,Rec). The dealing algorithm Deal(n, σ) takes as input
the number of shares to be generated nalong with the secret σand outputs shares σ1, . . . , σn. The
reconstruction algorithm Rec takes as input shares σ1, . . . , σnand outputs the secret σas long as
no more than tshares are corrupted (unavailable shares are set to ⊥and considered corrupted).
Schoenmakers [39] developed a simple VSS scheme based on discrete logarithms suitable for our
purposes.
Constructing Protocol πDLS.The main problem to be solved when realizing FD,F
DLS with a
protocol run by the stakeholders is that of generating uniform randomness for the leader selection
process while tolerating adversaries that may try to interfere by aborting or feeding incorrect
information to parties. In order to generate uniform randomness ρjfor epoch ej,j≥2, the
36

G
epoch starts new epoch
commit stage reveal stage
stake update
Figure 12: The two stages of the protocol πDPoS that use the blockchain as a broadcast channel.
elected stakeholders for epoch ej−1will employ a coin tossing scheme for which all honest parties
are guaranteed to receive output as long as there is an honest majority. The protocol has two
stages, commit and reveal which are split into phases. The stages of the protocol are presented in
Figure 12. The Commitment Phase covers the whole commitment stage, and proceeds as follows:
for 1 ≤i≤n, stakeholder Uisamples a uniformly random string ui∈ {0,1}Rlog τand randomness
rifor the underlying commitment scheme, generates shares σi
1, . . . , σi
n, and posts Com(ri, ui) to
the blockchain together with the encryptions of the all the shares under the public-key of each
respective shareholder. After 4kslots, players remove the kmost recent blocks of their chain, and
if commitments from a majority of stakeholders are posted on the blockchain and shares from a
majority of stakeholders have been received, the reveal stage starts (in the other case the protocol
halts). In the reveal stage there are two phase: the Reveal Phase and the Recovery Phase. In the
reveal phase, for 1 ≤i≤n, stakeholder Uiposts Open(ri, ui) to the blockchain. After 4kslots
players remove the most recent kblocks and identify all stakeholders that have issued openings
of the form Open(ri, ui). In the final Recovery Phase, lasting 2kslots, if a stakeholder Uathat
initially submitted a commitment is identified as not posting an opening to its commitment, the
honest parties can post all shares σa
1, . . . , σa
nin order to use Rec(σa
1, . . . , σa
n) to reconstruct ua.
Finally, each stakeholder uses the values uiobtained in the second round to compute ρj=Piui.
Protocol πDLS is described in Figure 13. We remark that it is possible to run the reveal and recovery
phases in parallel, however for improved efficiency we choose to run them sequentially.
5.3 Robust Transaction Ledger
We are now ready to state the main result of the section that establishes that the πDPOS protocol
with the protocol πDLS as a sub-routine implements a robust transaction ledger under the environ-
mental conditions that we have assumed. Recall that in the dynamic stake case we have to ensure
that the adversary cannot exploit the way stake changes over time and corrupt a set of stakeholders
that will enable the control of the majority of an elected committee of stakeholders in an epoch. In
order to capture this dependency on stake “shifts”, we introduce the following property.
Definition 5.1. Consider two slots sl1, sl2and an execution E. The stake shift between sl1, sl2is
the maximum possible statistical distance of the two weighted-by-stake distributions that are defined
using the stake reflected in the chain C1of some honest stakeholder active at sl1and the chain C2
of some honest stakeholder active at sl2respectively.
Given the definition above we can now state the following theorem.
Theorem 5.2. Fix parameters k, R, L ∈N, , σ ∈(0,1). Let R= 10kbe the epoch length and L
the total lifetime of the system. Assume the adversary is restricted to 1−
2−σrelative stake and
that the πSPOS protocol satisfies the common prefix property with parameters R, k and probability of
error CP, the chain quality property with parameters µ≥1/k, k and probability of error CQ and
37

Protocol πDLS
πDLS is a protocol run by a subset of elected stakeholders each one corresponding to a slot during an
epoch ejthat lasts R= 10kslots, without loss of generality denoted by U1, . . . , UR(which are not
necessarily distinct), and entails the following phases.
1. Commitment Phase (4kslots) When epoch ejstarts, for 1 ≤i≤n, stakeholder Uisamples a
uniformly random string uiand randomness rifor the underlying commitment scheme, generates
shares σi
1, . . . , σi
n←Deal(n, ui) and encrypts each share σi
kunder stakeholder Uk’s public-key.
Finally, Uiposts the encrypted shares and commitments Com(ri, ui) to the blockchain.
2. Reveal Phase (4kslots) After slot 4k, for 1 ≤i≤n, stakeholder Uiopens its commitment by
posting Open(ri, ui) to the blockchain provided that the blockchain contain valid shares from the
majority of U1, . . . , UR; if not, each Uiterminates.
3. Recovery Phase (2kslots) After slot 8k, for any stakeholder Uathat has not participated in
the reveal phase, i.e., it has not posted in Cdkan Open(ra, ua) message, for 1 ≤i≤R,Ui
submits its share σa
ifor insertion to the blockchain. When all shares σa
1, . . . , σa
nare available,
each stakeholder Uican compute Rec(σa
1, . . . , σa
n) to reconstruct ua(independently of whether Ua
opens the commitment or not).
The simulation of epochrnd req is then as follows.
•Given input (genblock req, Ui, ej,Sj), the stakeholder uses the commitment values in the
blockchain to compute ρj=Pl∈Lulwhere Lis the subset of stakeholders that were elected
in epoch ej. It returns (genblock, B0,Sj) with B0= (Sj, ρj).
Figure 13: Protocol πDLS.
the chain growth property with parameters τ≥1/2, k and probability of error CG. Furthermore,
assume that πDLS simulates a perfect beacon with distinguishing advantage DLS.
Then, the πDPOS protocol satisfies persistence with parameters kand liveness with parameters
u= 2kthroughout a period of Lslots (or Bad1/2happens) with probability 1−(L/R)(CQ+CP +CG +
DLS), assuming that σis the maximum stake shift over 10kslots, corruption delay D≥2R−4k
and no honest player is offline for more than kslots.
Proof. (sketch) Let us first consider the execution of πDPOS when FD,F
DLS is used instead of πDLS. Let
BADrbe the event that any of the three properties CP,CQ,CG is violated at round r≥1 while no
violation of any of them occurred prior to r. It is easy to see that Pr[∪r≤RBADr]≤CQ +CP +CG.
Conditioning now on the negation of this event, we can repeat the argument for the second epoch,
since D≥Rand thus the adversary cannot influence the stakeholder selection for the second epoch.
It follows that Pr[∪r≤LBADr]≤(L/R)(CQ +CP +CG). It is easy now to see that persistence
and liveness hold conditioning on the negation of the above event: a violation of persistence would
violate common prefix. On the other hand, a violation of liveness would violate either chain growth
or chain quality for the stated parameters.
Observe that the above result will continue to hold even if FD,F
DLS was weakened to allow the
adversary access to the random value of the next epoch 6kslots ahead of the end of the epoch.
This is because the corruption delay D≥2R−4k= 16k.
Finally, we examine what happens when FD,F
DLS is substituted by FD,F
LS and the execution of
protocol πDLS. Consider an execution with environment Zand adversary Aand event BAD that
happens with some probability βin this execution. We construct an adversary A∗that operates in
an execution with FD,F
DLS, weakened as in the previous paragraph, and induces the event BAD with
roughly the same probability β.A∗would operate as follows: in the first 4kslots, it will use an
honest party to insert in the blockchain the simulated commitments of the honest parties; this is
38

feasible for A∗as in 4kslots, chain growth will result in the blockchain growing by at least 2kblocks
and thus in the first kblocks there will be at least a single honest block included. Now A∗will
obtain from FD,F
DLS the value of the beacon and it will simulate the opening of all the commitments
on behalf of the honest parties. Finally, in the last 2kslots it will perform the forced opening of
all the adversarial commitments that were not opened. The protocol simulation will be repeated
for each epoch and the statement of the theorem follows.
Remark 2. We note that it is easy to extend the adversarial model to include fail-stop (and recover)
corruptions in addition to Byzantine corruptions. The advantage of this mixed corruption setting,
is that it is feasible to prove that we can tolerate a large number of fail-stop corruptions (arbitrarily
above 50%). The intuition behind this is simple: the forkable string analysis still applies even if
an arbitrary percentage of slot leaders is rendered inactive. The only necessary provision for this
would be expand the parameter kinverse proportionally to the rate of non-stopped parties. We omit
further details.
6 Anonymous Communication and Stronger Adversaries
The protocols constructed in the previous section are proven secure against delayed adaptive cor-
ruptions, meaning that, after requesting to corrupt a given party Ui, the adversary has to wait
for Dslots before the corruption actually happens. However it is desirable to make Das small as
possible, or even eliminate it altogether to achieve security against a standard adaptive adversary.
The delay is required because the adversary must not be able to corrupt parties once it knows
that they are the slot leaders for a given slot. However, notice that the slot leaders are selected by
weighting public keys by stake, while the adversary can only choose to corrupt a user Uiwithout
knowing its public key. Thus, the adversary must be able to observe communication between Ui
and the Diffuse functionality in order to determine which public key is associated with user Ui
and detect when Uiis selected as a slot leader. We will show that we can eliminate the delay
by extending our model with a sender anonymous broadcast channel (provided by the Diffuse
functionality) and having the environment activate all parties in every round. We introduce the
following modifications in the ideal functionalities:
•Diffuse Functionality: The functionality will work as described in Section 2 except that it
will remove all information about the sender Usof every message before delivering it to the
receiver Ur’s inbox (input tape), thus ensuring that the sender remain anonymous.7
•Key and Transaction Functionality: The functionality will work as described in Section 2 ex-
cept that it will allow immediate corruption of a user Uupon receiving a message (Corrupt, U)
from the adversary.
Apart from these modifications in the ideal functionalities, we also change the environment
behavior by requiring that it activates all users at every slot slj. Having all parties being activated
at every slot results in an anonymity set of size equal to the number of honest parties, making it
difficult for the adversary to associate a given public key with a user (i.e. any of the honest parties
could be associated with a given public key that is not associated with a corrupted party). In this
extended model we can reprove Theorem 5.2 without a delay Dby strengthening the restrictions
that are imposed on the environment in the following way.
7In practice, a sender anonymous broadcast channel with properties akin to those of the Diffuse functionality can
be implemented by Mix-networks [15] or DC-networks [16] that can be executed by the nodes running the protocol.
39

•We will say the adversary is restricted to less than 50% relative stake for windows of length D
if for all sets of consecutive slots of length D, the sum over all corrupted keys of the maximum
stake held by each key during this period of Dslots (in any possible Sj(r) where Ujis an
honest party) is no more than 50% of the minimum total stake during this period. In case
the above is violated an event Bad1/2
Dbecomes true for the given execution.
Using the above strengthened condition, we can remove the corruption delay requirement Din
Theorem 5.2 by assuming that Bad1/2is substituted with Bad1/2
D.
7 Incentives
So far our analysis has focused on the cryptographic adversary setting where a set of honest players
operate in the presence of an adversary. In this section we consider the setting of a coalition of
rational players and their incentives to deviate from honest protocol operation.
7.1 Input Endorsers
In order to address incentives, we modify further our basic protocol to assign two different roles
to stakeholders. As before in each epoch there is a set of elected stakeholders that runs the secure
multiparty coin flipping protocol and are the slot leaders of the epoch. Together with those there
is a (not necessarily disjoint) set of stakeholders called the endorsers. Now each slot has two
types of stakeholders associated with it; the slot leader who will issue the block as before and
the slot endorser who will endorse the input to be included in the block. Moreover, contrary
to slot leaders, we can elect multiple slot endorsers for each slot, nevertheless, without loss of
generality we just assume a single input endorser per slot in this description. While this seems like
an insignificant modification it gives us a room for improvement because of the following reason:
endorsers’ contributions will be acceptable even if they are dslots late, where d∈Nis a parameter.
Note that in case no valid endorser input is available when the slot leader is about to issue the
block, the leader will go ahead and issue an empty block, i.e., a block without any actual inputs
(e.g., transactions in the case of a transaction ledger). Note that slot endorsers just like slot leaders
are selected by weighing by stake and thus they are a representative sample of the stakeholder
population. In the case of a transaction ledger the same transaction might be included by many
input endorsers simultaneously. In case that a transaction is multiply present in the blockchain
its first occurrence only will be its “canonical” position in the legder. The enhanced protocol,
πDPOSwE, can be easily seen to have the same persistence and liveness behaviour as πDPOS: the
modification with endorsers does not provide any possibility for the adversary to prevent the chain
from growing, accepting inputs, or being consistent. However, if we measure chain quality in terms
of number of endorsed inputs included this produces a more favorable result: it is easy to see that
the number of endorsed inputs originating from a set of stakeholders Sin any k-long portion of the
chain is proportional to the relative stake of Swith high probability. This stems from the fact that
it is sufficient that a single honest block is created for all the endorsed inputs of the last dslots to
be included in it. Assuming d≥2k, any set of stakeholders Swill be an endorser in a subset of
the dslots with probability proportional to its cumulative stake, and thus the result follows.
As in bitcoin, stakeholders that issue blocks are incentivized to participate in the protocol
by collecting transaction fees. Contrary to bitcoin, of course, one does not need to incentivize
stakeholders to invest computational resources to issue blocks. Rather, availability and transaction
verification should be incentivized. Nevertheless, they have to be incentivized to be online often.
Any stakeholder, at minimum, must be online and operational in the following circumstances.
40

•In the slot prior to a slot she is the elected shareholder so that she queries the network and
obtains the currently longest blockchain as well as any endorsed inputs to include in the block.
•In the slot during which she is the elected shareholder so that she issues the block containing
the endorsed inputs.
•In a slot during the commit stage of an epoch where she is supposed to issue the VSS
commitment of her random string.
•In a slot during the reveal stage of an epoch where she is supposed to issue the required
opening shares as well as the opening to her commitment.
•In general, in sufficient frequency, to check whether she is an elected shareholder for the next
or current epoch.
•In a slot during which she is the elected input endorser so that she issues the endorsed input
(e.g., the set of transactions) that requires processing all available transactions and verifying
them.
In order to incentivize the above actions in the setting of a transaction ledger, fees can be
collected from those that issue transactions to be included in the ledger which can then be transfered
to the block issuers. In bitcoin, for instance, fees can be collected by the miner that produces a
block of transactions as a reward. In our setting, similarly, a reward can be given to the parties
that are issuing blocks and endorsing inputs. The reward mechanism does not have to be block
dependent as advocated in [34]. In our setting, it is possible to collect all fees of transactions
included in a sequence of blocks in a pool and then distribute that pool to all shareholders that
participated during these slots. For example, all input endorsers that were active may receive reward
proportional to the number of inputs they endorsed during a period of rounds (independently of
the actual number of transactions they endorsed). Other ways to distribute transaction fees are
also feasible (including the one that is used by bitcoin itself—even though the bitcoin method is
known to be vulnerable to attacks, e.g., the selfing-mining attack).
The reward mechanism that we will pair with input endorsers operates as follows. First we
set the endorsing acceptance window, dto be d= 2k. Let Cbe a chain consisting of blocks
B0, B1, . . .. Consider the sequence of blocks that cover the j-th epoch denoted by B1, . . . , Bswith
timestamps in {jR + 1,...,(j+ 1)R+ 2k}that contain an r≥0 sequence of endorsed inputs that
originate from the j-th epoch (some of them may be included as part of the j+ 1 epoch). We
define the total reward pool PRto be equal to the sum of the transaction fees that are included
in the endorsed inputs that correspond to the j-th epoch. If a transaction occurs multiple times
(as part of different endorsed inputs) or even in conflicting versions, only the first occurrence of
the transaction is taken into account (and is considered to be part of the ledger at that position)
in the calculation of P, where the total order used is induced by the order the endorsed inputs
that are included in C. In the sequence of these blocks, we identify by L1, . . . , LRthe slot leaders
corresponding to the slots of the epoch and by E1, . . . , Erthe input endorsers that contributed the
sequence of rendorsed inputs. Subsequently, the i-th stakeholder Uican claim a reward up to the
amount (β·|{j|Ui=Ej}|/r + (1 −β)·|{j|Ui=Lj}|/R)Pwhere β∈[0,1]. Claiming a reward is
performed by issuing a “coinbase” type of transaction at any point after 4kblocks in a subsequent
epoch to the one that a reward is being claimed from.
Observe that the above reward mechanism has the following features: (i) it rewards elected
committee members for just being committee members, independently of whether they issued a
41

block or not, (ii) it rewards the input endorsers with the inputs that they have contributed. (iii) it
rewards entities for epoch j, after slot jR + 4k.
We proceed to show that our system is a δ-Nash (approximate) equilibrium, cf. [31, Section
2.6.6]. Specifically, the theorem states that any coalition deviating from the protocol can add at
most an additive δto its total rewards.
A technical difficulty in the above formulation is that the number of players, their relative stake,
as well as the rewards they receive are based on the transactions that are generated in the course of
the protocol execution itself. To simplify the analysis we will consider a setting where the number
of players is static, the stake they possess does not shift over time and the protocol has negligible
cost to be executed. We observe that the total rewards (and hence also utility by our assumption
on protocol costs) that any coalition Vof honest players are able extract from the execution lasting
L=tR + 4k+ 1 slots, is equal to
RV(E) =
t
X
j=1
P(j)
all βIEj
V(E)
R+ (1 −β)SLj
V(E)
rj!
for any execution Ewhere common prefix holds with parameter k, where rjis the total endorsed
inputs emitted in the j-th epoch (and possibly included at any time up to the first 2kslots of epoch
j+ 1), P(j)
all is the reward pool of epoch j,SLj
V(E) is the number of times a member of Vwas
elected to be a slot leader in epoch jand IEj
V(E) the number of times a member of Vwas selected
to endorse an input in epoch j.
Observe that the actual rewards obtained by a set of rational players Vin an execution Emight
be different from RV(E); for instance, the coalition of Vmay never endorse a set of inputs in which
case they will obtain a smaller number of rewards. Furthermore, observe that we leave the value
of RV(E) undefined when Eis an execution where common prefix fails: it will not make sense
to consider this value for such executions since the view of the protocol of honest parties can be
divergent; nevertheless this will not affect our overall analysis since such executions will happen
with sufficiently small probability.
We will establish the fact that our protocol is a δ-Nash equilibrium by proving that the coalition
V, even deviating from the proper protocol behavior, it cannot obtain utility that exceeds RV(E)+δ
for some suitable constant δ > 0.
Theorem 7.1. Fix any δ > 0; the honest strategy in the protocol is a δ-Nash equilibrium against any
coalition commanding a proportion of stake less than (1 −)/2−σfor some constants , σ ∈(0,1)
as in Theorem 5.2, provided that the maximum total rewards Pall provided in all possible protocol
executions is bounded by a polynomial in λ, while CQ +CP +CG +DLS is negligible in λ.
Proof sketch. Consider a coalition of rational players Vrestricted as in the statement of the the-
orem, that engages in a protocol execution together with a number of other players that follow
the protocol faithfully for a total number of Lepochs. We will show that any deviation from the
protocol will not result in substantially higher rewards for V. Observe that based on Theorem 5.2,
no matter the strategy of V, with probability 1 −(L/R)(CQ +CP +CG ) the protocol will enable
all users to obtain the rewards they are entitled to as slot leaders and input endorsers. The latter
stems from the following. First, from persistence and liveness, at least one honest block will be
included every kblocks and hence, in each epoch, all input endorsers that follow the protocol will
have the opportunity to act as input endorsers as many times they were elected to be. Second,
the rewards received will be proportional to the times each party is an input endorser and issued a
block successfully as well as equal to the number of times it is a slot leader. We observe that except
with probability (L/R)(CQ +CP +CG) the utility received by coalition Vis equal to RV. It follows
42

that player Vhas expected utility at most E[RV]+(L/R)(CQ +CP +CG)PAll, where PAll is the
maximum amount of rewards produced in the lifetime of all possible executions. The result follows
by the assumption in the statement of the theorem since (L/R)(CQ +CP +CG)PAll ≤δ.
Remark 3. In the above theorem, for simplicity, we assumed that protocol costs are not affecting
the final utility (in essence this means that protocol costs are assumed to be negligible). Nevertheless,
it is straightforward to extend the proof to cover a setting where a negative term is introduced in
the payoff function for each player proportional to the number of times inputs are endorsed and
the number of messages transmitted for the MPC protocol. The proof would be resilient to these
modifications because endorsed inputs and MPC protocol messages cannot be stifled by the adversary
and hence the reward function can be designed with suitable weights for such actions that offsets
their cost. Stil l note that the rewards provided are assumed to be “flat” for both slots and endorsed
inputs and thus the costs would also have to be flat. We leave for future work the investigation of
a more refined setting where costs and rewards are proportional to the actual computational steps
needed to verify transactions and issue blocks.
Remark 4. The reward function described, only considers the number of times an entity was an
input endorser without considering the amount of work that was put to verify the given transactions.
Furthermore it is not sensitive to whether a slot leader issued a block or not in its assigned time slot.
We next provide some context behind these choices. First suppose that slot leaders do not receive
a reward when they do not issue a block. It is easy to see that when all parties follow the protocol
the parties will receive the proportion from the reward pool that is associated to block issuance
roughly proportional to their stake. Nevertheless, a malicious coalition can easily increase the ratio
of these rewards by performing a block witholding attack (in this case this would amount to a selfish
mining attack). Given that this happens with non-negligible probability a straightforward definition
of RV(E)that respects this assignment is vulnerable to attack and hence a δ-Nash equilibrium
theorem cannot be shown. Next, we consider the case of extending the reward function so that input
endorsers that are rewarded based on the transactions they verify (as opposed to the flat reward
we considered in the above theorem). Special care is necessary to design this function. Indeed the
straightforward way to implement it, which is if the first input endorser to verify a transaction that
is part of the pool can make a higher claim for its fee, then there is a strategy for an adversary
to deviate from the protocol and improve its ratio of rewards: perform block withholding and/or
endorsed input censorship to remove endorsed inputs from the blockchain that originate to honest
parties. Then include the removed transactions in endorsed input that will be transmitted in the
last possible opportunity. As before, given the attack, the natural way to define RV(E)is susceptible
to it and hence a δ-Nash equilibrium theorem cannot be shown.
A possible direction for ameliorating the problem raised in Remark 4 above, is to share the
transaction fee between all the input endorsers that endorsed it. This suggests the following mod-
ification to the protocol: whenever you are an input endorser you should attempt to include all
transactions that you have collected for a sequence of kslots and retransmit your endorsed input in
case it is removed from the main chain. We leave the analysis of such class of reward mechanisms
for future work.
8 Stake Delegation
As discussed in the previous section, stakeholders must be online in order to generate blocks when
they are selected as slot leaders. However, this might be unattractive to stakeholders with a small
stake in the system. Moreover, requiring that a majority of elected stakeholders participate in the
43

coin tossing protocol for refreshing randomness introduces a strain on the on the stakeholders and
the network, since it might require broadcasting and storing a large number of commitments and
shares.
We mitigate these issues by providing a method for reducing the size of the group of stakehold-
ers that engage in the coin tossing protocol. Instead of the elected stakeholders directly forming
the committee that will run coin tossing, a group of delegates will act on their behalf. In more
detail, we put forth a delegation scheme, whereby stakeholders will authorize other entities, called
delegates, who may be stakeholders themselves, to represent them in the coin tossing protocol. A
delegate may participate in the protocol only if it represents a certain number of stakeholders whose
aggregate stake exceeds a given threshold. Such a participation threshold ensures that a “fragmen-
tation” attack, that aims to increase the delegate population in order to hurt the performance of
the protocol, cannot incur a large penalty as it is capable to force the size of the committee that
runs the protocol to be small (it is worth noting that the delegation mechanism is similar to mining
pools in proof-of-work blockchain protocols).
8.1 Minimum Committee Size
To appreciate the benefits of delegation, recall that in the basic protocol (πDPoS) a committee
member selected by weighing by stake is honest with probability 1/2 + (this being the fraction
of the stake held by honest players). Thus, the number of honest players selected by kinvocations
of weighing by stake is a binomial distribution. We are interested in the probability of a malicious
majority, which can be directly controlled by a Chernoff bound. Specifically, if we let Ybe the
number of times that a malicious committee member is elected then
Pr[Y≥k/2] = Pr[Y≥(1 + δ)(1/2−)k]
≤exp(−min{δ2, δ}(1/2−)k/4)
<exp(−δ2(1/2−)k/4)
for δ= 2/(1 −2). Assuming  < 1/4, it follows that δ < 1.
Consider the case that = 0.05; then we have the bound exp(−0.00138 ·k) which provides
an error of 1/1000 as long as k≥5000. Similarly, in the case = 0.1, we have the bound
exp(−0.00625k) which provides the same error for k≥1100.
We observe that in order to withstand a significant number of epochs, say 215 (which, if we
equate a period with one day, will be 88 years), and require error probability 2−40, we need that
k≥32648.
In cases where the wealth in the system is not concentrated among a small set of stakeholders
the above choice is bound to create a very large committee. (Of course, the maximum size of the
committee is k.)
8.2 Delegation Scheme.
The concept of delegation is simple: any stakeholder can allow a delegate to generate blocks on her
behalf. In the context of our protocol, where a slot leader signs the block it generates for a certain
slot, such a scheme can be implemented in a straightforward way based on proxy signatures [10].
A stakeholder can transfer the right to generate blocks by creating a proxy signing key that
allows the delegate to sign messages of the form (st, d, slj) (i.e., the format of messages signed in
Protocol πDPoS to authenticate a block). In order to limit the delegate’s block generation power
to a certain range of epochs/slots, the stakeholder can limit the proxy signing key’s valid message
space to strings ending with a slot number sljwithin a specific range of values. The delegate
44

can use a proxy signing key from a given stakeholder to simply run Protocol πDPoS on her behalf,
signing the blocks this stakeholder was elected to generate with the proxy signing key. This simple
scheme is secure due to the Verifiability and Prevention of Misuse properties of proxy signature
schemes, which ensure that any stakeholder can verify that a proxy signing key was actually issued
by a specific stakeholder to a specific delegate and that the delegate can only use these keys to
sign messages inside the key’s valid message space, respectively. We remark that while proxy
signatures can be described as a high level generic primitive, it is easy to construct such schemes
from standard digital signature schemes through delegation-by-proxy as shown in [10]. In this
construction, a stakeholder signs a certificate specifying the delegates identity (e.g., its public key)
and the valid message space. Later on, the delegate can sign messages within the valid message
space by providing signatures for these messages under its own public key along with the signed
certificate. As an added advantage, proxy signature schemes can also be built from aggregate
signatures in such a way that signatures generated under a proxy signing key have essentially the
same size as regular signatures [10].
An important consideration in the above setting is the fact that a stakeholder may want to
withdraw her support to a stakeholder prior to its proxy signing key expiration. Observe that proxy
signing keys can be uniquely identified and thus they may be revoked by a certificate revocation
list within the blockchain.
8.2.1 Eligibility threshold
Delegation as described above can ameliorate fragmentation that may occur in the stake distribu-
tion. Nevertheless, this does not prevent a malicious stakeholder from dividing its stake to multiple
accounts and, by refraining from delegation, induce a very large committee size. To address this,
as mentioned above, a threshold T, say 1%, may be applied. This means that any delegate repre-
senting less a fraction less than Tof the total stake is automatically barred from being a committee
member. This can be facilitated by redistributing the voting rights of delegates representing less
than Tto other delegates in a deterministic fashion (e.g., starting from those with the highest stake
and breaking ties according to lexicographic order). Suppose that a committee has been formed,
C1, . . . , Cm, from a total of kdraws of weighing by stake. Each committee member will hold ki
such votes where Pm
i=1 ki=k. Based on the eligibility threshold above it follows that m≤T−1
(the maximum value is the case when all stake is distributed in T−1delegates each holding Tof
the stake).
9 Attacks Discussion
We next discuss a number of practical attacks and indicate how they are reflected by our modeling
and mitigated.
Double spending attacks In a double spending attack, the adversary wishes to revert a trans-
action that is confirmed by the network. The objective of the attack is to issue a transaction, e.g.,
a payment from an adversarial account holder to a victim recipient, have the transaction confirmed
and then revert the transaction by, e.g., including in the ledger a second conflicting transaction.
Such an attack is not feasible under the conditions of Theorem 5.2. Indeed, persistence ensures
that once the transaction is confirmed by an honest player, all other honest players from that point
on will never disagree regarding this transaction. Thus it will be impossible to bring the system to
a state where the confirmed transaction is invalidated (assuming all preconditions of the theorem
hold). See the next section for an experimental discussion about double spending.
45

Grinding attacks In stake grinding attacks, the adversary tries to influence the slot leader
selection process to improve its chances of being selected to generate blocks (which can be used to
perform other attacks such as double spending). Basically, when generating a block that is taken
as input by the slot leader selection process, the adversary first tests several possible block headers
and content in order to find the one that gives it the best chance of being selected as a slot leader
again in the future. While this attack affects PoS based cryptocurrencies that collect randomness
for the slot leader selection process from raw data in the blockchain itself (i.e. from block headers
and content), our protocol uses a standard coin tossing protocol that is proven to generate unbiased
uniform randomness as discussed in Section 5.2. We show that an adversary cannot influence the
randomness generated in Figure 13, which is guaranteed to be uniformly random, thus guaranteeing
that slot leaders are selected with probability proportional to their stake.
Transaction denial attacks In a transaction denial attack, the adversary wishes to prevent a
certain transaction from becoming confirmed. For instance, the adversary may want to target a
specific account and prevent the account holder from issuing an outgoing transaction. Such an
attack is not feasible under the conditions of Theorem 5.2. Indeed, liveness ensures that, provided
the transaction is attempted to be inserted for a sufficient number of slots by the network, it will
be eventually confirmed.
Desynchronization attacks In a desynchronization attack, a shareholder behaves honestly but
is nevertheless incapable of synchronizing correctly with the rest of the network. This leads to
ill-timed issuing of blocks and being offline during periods when the shareholder is supposed to
participate. Such an attack can be mounted by preventing the party’s access to a time server or
any other mechanism that allows synchronization between parties. Moreover, a desynchronization
may also occur due to exceedingly long delays in message delivery. Our model allows parties to
become desynchronized by incorporating them into the adversary. No guarantees of liveness and
persistence are provided for desynchronized parties and thus we can get security as long as parties
with less than 50% of stake get desynchronized. If more than parties get desynchronized our
protocol can fail. More general models like partial synchrony [19, 35] are interesting to consider in
the PoS design setting.
Eclipse attacks In an eclipse attack, message delivery to a shareholder is violated due to a
subversion in the peer-to-peer message delivery mechanism. As in the case of desynchronization
attacks, our model allows parties to be eclipse attacked by incorporating them into the adversary.
No guarantees of liveness or persistence are provided for such parties.
51% attacks A 51% attack occurs whenever the adversary controls more than the majority of
the stake in the system. It is easy to see that any sequence of slots in such a case is with very high
probability forkable and thus once the system finds itself in such setting the honest stakeholders
may be placed in different forks for long periods of time. Both persistence and liveness can be
violated.
Bribery Attacks In bribery attacks [11], an adversary deliberately pays miners (through cryp-
tocurrencies or fiat money) to work on specific blocks and forks, aiming at generating an arbitrary
fork that benefits the adversary (e.g. by supporting a double spending attack). Miners of PoW
based cryptocurrencies do not have to own any stake in order to mine blocks, which makes this
attack strategy feasible. In this setting, if the adversary offers a bribe higher than the reward
46

for correctly generating a block, any rational miner has a clear incentive to accept the bribe and
participate in the attack since it increases the miner’s financial outcome. However, in our PoS
based protocol, malicious slot leaders who agree to deliberately attack the system not only risk to
forego any potential profit they would earn from behaving honestly but may also risk to lose equity.
Notice that slot leaders must have money invested in the system in order to be able to generate
blocks and if an attack against the system is observed it might bring currency value down. Even
if the bribe is higher than the reward for correct behavior, the loss from currency devaluation can
easily offset any additional profits made by participating in this attack. Hence, bribery attacks may
be be less effective against a PoS based consensus protocol than a PoW based one. Currently our
rationality model does not formally encompass this attack strategy and investigating its efficacy
against PoS based consensus protocols is left as a future work.
Long-range attacks An attacker who wishes to double spend at a later point in time can mount
a long-range attack [12] by computing a longer valid chain that starts right after the genesis block
where it is the single stakeholder actively participating in the protocol. Even if this attacker
owns a small fraction of the total stake, it can locally compute this chain generating only the
blocks for slots where it is elected the slot leader and keep generating blocks ahead of current
time until its alternative chain has more blocks than the main chain. Now, the attacker can post
a transaction to the main chain, wait for it to be confirmed (and for goods to be delivered in
exchange for the transaction) and present the longer alternative chain to invalidate its previously
confirmed transaction. This attack is ineffective against Ouroboros for two reasons: Protocol πDLS
will only output valid leader selection data allowing for the protocol to continue if a majority of
the stakeholders participate (or have delegates participate on their behalf) and stakeholders will
reject blocks generated for slots that are far ahead of time. Since the alternative chain is generated
artificially with blocks and protocol messages generated solely by an attacker who controls a small
fraction of the stake the leader selection data needed to start new epochs will be considered invalid
by other nodes. Even if the attacker could find a strategy to generate an alternative chain with
valid leader selection data, presenting this chain and its blocks generated at slots that are far ahead
of time would not result in a successful attack since those blocks far ahead of time would be rejected
by the honest stakeholders and the final alternative chain would be shorter than the main chain.
Nothing at stake attacks The “nothing at stake” problem refers in general to attacks against
PoS blockchain systems that are facilitated by shareholders continuing simultaneously multiple
blockchains exploiting the fact that little computational effort is needed to build a PoS blockchain.
Provided that stakeholders are frequently online, nothing at stake is taken care of by our analysis
of forkable strings (even if the adversary brute-forces all possible strategies to fork the evolving
blockchain in the near future, there is none that is viable), and our chain selection rule that
instructs players to ignore very deep forks that deviate from the block they received the last time
they were online. It is also worth noting that, contrary to PoW-based blockchains, in our protocol
it is infeasible to have a fork generated in earnest by two shareholders. This is because slots are
uniquely assigned and thus at any given moment there is a single uniquely identified shareholder
that is elected to advance the blockchain. Players following the longest chain rule will adopt the
newly minted block (unless the adversary presents at that moment an alternative blockchain using
older blocks). It is remarked in [13] that the “tragedy of commons” might lead stakeholders in
some PoS based schemes to adhere to attacks because they do not have the power to deter attacks
by themselves and would incur financial losses even if they did not join the attack. This would
lead rational stakeholders to accept small bribes in alternative currencies that might at least obtain
47

some financial gain. However, in the incentive structure of Ouroboros, slot leaders and endorsers
who could potentially join an attack would receive rewards in both the main and the adversarial
chain, resulting in those stakeholders not achieving higher profits by joining the attack.
Past majority attacks As stake moves our assumption is that only the current majority of
stakeholders is honest. This means that past account keys (which potentially do not hold any stake
at present) may be compromised. This leads to a potential vulnerability for any PoS system since a
set of malicious shareholders from the past can build an alternative blockchain exploiting such old
accounts and the fact that it is effortless to build such a blockchain. In light of Theorem 5.2 such
attack can only occur against shareholders who are not frequently online to observe the evolution
of the system or in case the stake shifts are higher than what is anticipated by the preconditions
of the theorem. This can be seen a special instance of the nothing at stake problem, where the
attacker no longer owns any stake in the system and is thus free from any financial losses when
conducting the attack.
Selfish-mining In this type of attack, an attacker withholds blocks and releases them strategi-
cally attempting to drop honestly generated blocks from the main chain. In this way the attacker
reduces chain growth and increases the relative ratio of adversarially generated blocks. In conven-
tional reward schemes, as that of bitcoin, this has serious implications as it enables the attacker
to obtain a higher rate of rewards compared to the rewards it would be receiving in case it was
following the honest strategy. Using our reward mechanism however, selfish mining attacks are
neutralized. The intuition behind this, is that input endorsers, who are the entities that receive
rewards proportionally to their contributions, cannot be stifled because of block withholding: any
input endorser can have its contribution accepted for a sufficiently long period of time after its
endorsement took place, thus ensuring it will be incorporated into the blockchain (due to sufficient
chain quality and chain growth). Given that input endorsers’ contributions are (approximately)
proportional to their stake this ensures that reward distribution cannot be affected substantially
by block withholding.
10 Experimental Results
We have implemented a prototype instantiation of Ouroboros in Haskell as well as in the Rust-
based Parity Ethereum client in order to evaluate its concrete performance. More specifically, we
have implemented Protocol πDPoS using Protocol πDLS to generate leader selection parameters (i.e.,
generating fresh randomness for the weighed stake sampling procedure). For this instantiation, we
use the PVSS scheme of [39] implemented over the elliptic curve secp256r1. This PVSS scheme’s
share verification information includes a commitment to the secret, which is also used as the
commitment specified in protocol πDLS; this eliminates the need for a separate commitment to
be generated and stored in the blockchain. In order to obtain better efficiency, the final output ρ
of Protocol πDLS is a uniformly random binary string of 32 bytes. This string is then used as a
seed for a PRG (ChaCha in our implementation, [8]) and stretched into Rrandom labels of log τ
bits corresponding to each slot in an epoch. The weighing by stake leader selection process is then
implemented by using the random binary string associated to each epoch to perform the sequence
of coin-flips for selecting a stakeholder. The signature scheme used for signing blocks is ECDSA,
also implemented over curve secp256r1.
48

10.1 Transaction Confirmation Time Under Optimal Network Conditions
We first examine the time required for confirming a transaction in a setting where the network is
not under substantial load and transactions are processed as they appear.
Adversary BTC OB Covert OB General
0.10 50 3 5
0.15 80 5 8
0.20 110 7 12
0.25 150 11 18
0.30 240 18 31
0.35 410 34 60
0.40 890 78 148
0.45 3400 317 663
Figure 14: Transaction confirmation times in minutes that achieve assurance 99.9% against a hypo-
thetical double spending attack with different levels of adversarial power for Bitcoin and Ouroboros
(both covert and general adversaries).
In Fig. 14 we lay out a comparison in terms of transaction confirmation time between Bitcoin
and Ouroboros showing how much a verifier has to wait to be sure that the best possible8double-
spending attack succeeds with probability less than 0.1%. In the case of Bitcoin, we consider a
double-spending attacker that commands a certain percentage of total hashing power and wishes
to revert a transaction. The attacker attempts to double-spend via a block-witholding attack as
described in the same paper (the attacker mines a private fork and releases it when it is long
enough). In the case of Ouroboros we consider a double spending attacker that attempts to brute
force the space of all possible forks for the current slot leader distribution in a certain segment of
the protocol and commands a certain percentage of the total stake. We consider both the covert
and the general adversarial setting for Ouroboros.
In all of the scenarios, we measure the number of minutes that one has to wait in order to achieve
probability of double spending less than 0.1%. In Fig. 15 we present a graph that illustrates the
speedup graphically.
We note that the above measurements compare our Ouroboros implementation with Bitcoin
in the way the two systems are parameterized (with 10 minute block production rate for Bitcoin
and 20 second slots for Ouroboros, a conservative parameter selection). Exploring alternative
parameterizations for Bitcoin (such as making the proof-of-work easier) can speed up the transaction
processing, nevertheless this cannot be done without carefully measuring the impact on overall
security.
10.2 Absolute Performance of Ouroboros
We implemented Ouroboros as an instance of the Rust-based Ethereum Parity client.9Subse-
quently, experiments were run using Amazon’s Elastic Compute Cloud (EC2) ‘c4.2xlarge‘ instances
in the ‘us-east-1‘ region with a smaller “runner” instance responsible for coordinating each of the
“worker” instances.
Each experiment consists of several steps:
8The “best possible” is only in the the case of Ouroboros, for Bitcoin we use the best known attack.
9Ethcore - Parity. https://ethcore.io/parity.html
49

0.1 0.2 0.3 0.4
4
6
8
10
12
14
16
Adversarial Strength of Blockwitholding Attacker
Speedup OB/BTC
Confirmation time speed up of Ouroboros over BTC
Covert
General
Figure 15: Ouroboros vs. Bitcoin speedup of transaction confirmation time against a hypothetical
double spending attacker for assurance level 99.9%. Ouroboros is at least 10 to 5 times faster for
regular adversaries and 16 to 10 times faster for covert adversaries.
1. Each worker instance builds a clean Docker image containing a specific revision of our fork of
the Parity software10 containing the Ouroboros proof-of-concept changes based on the Parity
1.6.8 release.
2. Each worker instance is started in an “isolated” mode where none of the nodes talk to each
other. During this period, a Parity account is recovered on each node and a start time for
the network is established.
3. Each worker instance is restarted in a production mode that allows communication between
the nodes and transactions to be mined.
4. A single worker instance is informed about all the other nodes. All nodes become aware of
all other nodes via Parity’s peer-to-peer discovery methods.
5. Each worker instance has a number of transactions generated and ingested.
In each experiment, 650,000 total transactions are generated between the participating nodes
who shared stake equally. The amount transferred in any given transaction is small enough to
avoid any account running out of funds. Each instance generates all the transactions using a hard-
coded shared random seed, then keeps the transactions originating from the local user account. 20
transactions are saved in a single JSON file, ready to be directly passed to the Parity RPC endpoint
using the ‘curl‘ command line tool. During ingestion, a single file of 20 transactions is ingested
and one second is spent idle between each file to avoid overwhelming the instances with too many
requests.
Various setups were tested, focusing on adjusting the Ouroboros slot duration and the number
of participating nodes. 10, 20, 30, and 40 nodes were tested, ultimately limited by the number of
10Available from https://github.com/input-output-hk/parity/tree/experiment-2
(020fd77dc70d3f25e0e0f44bd6b1e19ccf3790d3)
50

Figure 16: Measuring transactions per second in a 40 node, equal stake deployment with slot
length of 5 seconds.
instances allowed in a single EC2 region. Slot durations of 5, 10, and 20 seconds were also tested.
Variance between experiments was small. In Figure 16 we present the case of 40 nodes and slot
length of 5 seconds that exhibits a median value of 257.6 transaction per second.
11 Acknowledgements
We thank Ioannis Konstantinou who contributed in a preliminary version of our protocol. We
thank Lars Br¨unjes, Duncan Coutts, Kawin Worrasangasilpa for comments on previous drafts of
the article. We thank Peter Gaˇzi for comments on previous drafts of the article and assisting us to
generalize Theorem 4.26 to viable forks. We thank George Agapov for the prototype implementation
of our protocol in Haskell and Jake Goulding for the Parity based implementation.
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