Anonymity in voting revisited
Hugo Jonker1,2and Wolter Pieters3,⋆
1Eindhoven University of Technology, The Netherlands
2University of Luxembourg, Luxembourg
Abstract. According to international law, anonymity of the voter is a
fundamental precondition for democratic elections. In electronic voting,
several aspects of voter anonymity have been identified. In this paper,
we re-examine anonymity with respect to voting, and generalise existing
notions of anonymity in e-voting. First, we identify and categorise the
types of attack that can be a threat to anonymity of the voter, including
different types of vote buying and coercion. This analysis leads to a
categorisation of anonymity in voting in terms of a) the strength of the
anonymity achieved and b) the extent of interaction between voter and
attacker. Some of the combinations, including weak and strong receipt-
freeness, are formalised in epistemic logic.
In the field of peer-to-peer (P2P) networks, much effort has been put into for-
malizing the concept of anonymity of messages (e.g. ). Intuitively, anonymity
means that it is impossible to determine who sent which message to whom. De-
pending on the context, different formalizations of the notion of anonymity seem
to be necessary .
The concept of anonymity is also of importance in electronic voting – often,
voters should have the ability to vote without anybody else knowing which op-
tion they voted for (although in some countries, such as the United Kingdom and
New Zealand, this is ultimately not the case). In the electronic voting commu-
nity, the property expressing precisely that is usually called “privacy” instead of
anonymity . In voting, however, enabling privacy is not sufficient, as this does
not prevent vote buying. To prevent vote buying, an election needs to require
privacy – no voter should be able to convince any other party of how she voted.
The concept of receipt-freeness expresses that a voter cannot convince any
other party of how she voted by creating a receipt. The notion has been intro-
duced by , after which various receipt-free voting protocols were proposed,
such as [9,17]. Delaune et al.  provide a definition of receipt-freeness based on
observational equivalence. Independently, Jonker and De Vink  provide an
alternate definition that allows identification of receipts. Juels et al. note in 
⋆Research was carried out partially at Radboud University, Nijmegen, The Nether-
lands, and supported by NWO, the Netherlands Organisation for Scientific Research.
that receipt-freeness is not sufficient to prevent coercion in electronic elections,
and they introduced the notion of coercion-resistance. This broader notion is
again formalized by Delaune et al. in .
Given the differences in approachesand in notions, the question arises whether
these notions capture the specific needs for anonymity in voting. The three main
levels of anonymity that have been identified in voting, capture progressively
more strict notions of anonymity. The notion of receipt-freeness was motivated
as necessary to provide secret-ballot elections. If receipts can be obtained, using
a voting booth makes no difference to the secrecy: Votes can be bought, and
voters can be coerced.
To address the question of whether or not the notion of receipt-freeness is
sufficient, we reexamine voter influencing, focusing on vote buying. What is vote
buying, when can an action be called vote buying and when is it an election
promise? As this is, ultimately, a subjective issue, the goal is not to provide a
yes-or-no test. Instead, we aim to arrive at a charactarisation of vote buying /
election promises, which will enable election officials to decide which practices
are allowed and which should be abolished. Based on these findings, we then
reexamine the concept of receipt-freeness and adapt it to encompass uncovered
Distinctions between vote buying and election promises have been investigated
by economists, philosophers and political scientists before.
Van Acker  discusses the relation between the notions of coercion, forced
abstention, randomisation and simulation. However, he includes vote buying in
the concept of coercion.
Kochin and Kochin  discuss the issue of giving benefits to individual voters
versus giving benefits to identifiable groups. They also consider the difference
between benefits offered through the normal processes of government (related
to being elected) versus benefits offered through private arrangements. Thirdly,
they mention that trading votes for or against proposals between parties or
members in parliament is acceptable.
The latter practice is also mentioned by Hasen  and called “legislative
logrolling”. Hasen further differentiates the issues of corporate vote buying, pay-
ments to increase turnout, campaign promises and campaign contributions, and
vote buying in so-called “special district”1elections.
Schaffer  distinguishes between instrumental, normative and coercive com-
pliance in relation to vote buying. Instrumental compliance covers tangible bene-
fits in exchange for votes, normative compliance means voting based on a feeling
of obligation, and coercive compliance denotes voting based on threats. Schaffer
also mentions the possibility that money is offered for not changing voting be-
haviour. In order to check compliance, a buyer may monitor the individual vote,
1“a special purpose unit of government assigned the performance of functions affecting
definable groups of constituents more than other constituents”
monitor the aggregate turnout, prevent people from voting altogether, make the
rewards dependent on his election, make voters believe in his goodness or make
voters feel personally obliged. The applicability of these strategies is dependent
on the mode of compliance the buyer is seeking. From the perspective of voters,
benefits can be received in the form of payment, gift or wage, with different
explicit and implicit meanings in terms of modes of compliance.
From these papers it is clear that what exactly constitutes acceptable influ-
ence and what does not, depends on the type of elections, the society in which
the elections are being held and the participants of the elections. In the end, the
matter is ultimately a subjective one. However, by determining the various char-
acteristics of vote buying, and their respective ranges, it is possible to establish
a pre-election consensus on allowed and disallowed practices. Such a pre-election
consensus enables putting precise requirements on voting systems to support the
one type of behaviour, while preventing the other type.
1.2 Outline of the paper
In Section 2, we identify and categorise the types of attack that can be a threat
to anonymity of the voter, including different types of vote buying and coer-
cion. This analysis leads to a categorisation of anonymity in voting in terms
of a) the strength of the anonymity achieved and b) the extent of interaction
between voter and attacker, which is presented in Section 3. In Section 4, some
of the combinations, including weak and strong receipt-freeness, are formalised
in epistemic logic. The last section presents conclusions and future work.
2Characteristics of voter influencing
In this section, we investigate the characteristics of voter influencing. The ex-
amples below are used as supporting guidelines throughout the section. These
examples are deliberately without context – in lieu of what was established in
Section 1.1. The reason for this is that the aim is to discover the generic char-
acteristics involved, irrespective of social and electoral context. The examples
are not meant to capture any precise attempt at influencing voters, but rather
they convey a broad idea of a, possibly controversial, attempt at changing the
outcome of an election by targeting the voters.
Example 1 (handout). At the polling station, I give each voter 100 euros together
with mentioning my candidacy.
Example 2 (theme park). The district with the highest percentage of votes for
me gets a theme park as my first act as elected official.
Example 3 (zalmsnip). If I get elected, everyone gets 100 euros tax refund.
Example 4 (election promise). If I get elected, disabled child prodigies get 100
euros (i.e. children with a physical handicap, who are members of Mensa).
Example 5 (rf). I give 100 euro for anyone voting for the Democratic Party.
Example 6 (non-rf). I give 100 euro for anyone not voting for the Democratic
Example 7 (reimburse). I provide a reimbursement for voters for time away from
The zalmsnip example is based on a tax rebate that occurred in the Nether-
lands in 1998 and 1999. The rf and non-rf examples are inspired by the notion
of receipt-freeness. The reimburse example is inspired by this practice occurring
in the 19th century in the Netherlands.
Note that – as long as there is no request to vote in a specific way – example 1
can be considered legitimate. Examples 3 and 4 are fabrications resembling pos-
sible election promises. Example 2 is dubious; examples 5–7 are outright illegal.
2.1Legal and illegal influencing
Influencing voters can be done either legally or illegally. To avoid a legal dis-
cussion on what is allowed by which laws, we only focus upon characterising
what is desirable. As established in the introduction, this is a subjective notion.
The aim here is to outline the range of possibilities available, indicate where the
boundary between desirable and undesirable lies and give a supportive reasoning
for where we feel this boundary lies.
Note that, in general, there are two methods to influence a voter’s vote:
coercion where voters are threatened to ensure compliance;
enticement where voters are seduced into compliance.
Whereas persuasion is allowed, buying and coercion are not. Both buying and
coercion require proof of compliance, whereas persuasion does not. Both buying
and persuasion are dependent on voluntary cooperation of the voter, coercion is
Voter influencing can be considered acceptable or unacceptable. What is
considered acceptable depends on culture and the nature of the elections. That
there can exist both acceptable and unacceptable variants of the above two
methods is illustrated by the following list.
– acceptable coercion claiming that all other candidates have significantly worse
plans for the voter
– unacceptable coercion threatening with physical violence in case of non-
– acceptable enticement promising to lower taxes
– unacceptable enticement paying a voter to vote for you
The above list clearly indicates, that there is a distinction between accept-
able influence and unacceptable influence. To establish the characteristics that
together determine the acceptability, we construct an objective tree of voter in-
fluencing in Section 2.2. Objective trees are attack trees (see [19,13]), but focus
upon meeting goals instead of achieving attacks.
Our objective tree deviates slightly from normal attack trees. The purpose
of our tree is to determine characteristics that distinguish acceptable from unac-
ceptable influence. To elucidate these detailed characteristics, details need to be
explicit in the tree. Hence, where normally attributes would be used, we promote
these characteristics to leaves. This makes these characteristics explicit.
2.2Classifying vote buying
Based on the literature, the examples and the analysis above, the tree in Figure 1
was constructed and dimensions of vote buying were clarified. The main goal in
the tree is to buy a vote, by means of persuasion. The tree is thus from the
perspective of a vote buyer. Where necessary, the range of possible values has
been indicated in the tree (as leaves).
– or reward time
• leaf before vote casting
• and later
∗ or trust required
· leaf rewarding sureness
· leaf consequences of non-reward
· leaf ensurance of compliance
∗ or hand out reward
· leaf after vote casting
· leaf after ballot box closes
· leaf after vote counting
– or type of reward
• leaf money
• leaf goods
• leaf immaterial
– or rewarding conditions
• leaf cast vote
• leaf election win
• leaf unconditional
• leaf complex
– leaf group size of reward receivers
– or proving compliance
• leaf before rewarding
• leaf after rewarding
• leaf not required
– leaf reward related to election
Fig.1. Objective tree for vote buying
The tree is to be read as follows: before each entry, the type of the entry is
marked (‘or’ for or-nodes, ‘and’ for and-nodes, ‘leaf’ for leaves). Or-nodes are
nodes of which at least one of the branches must be satisfied, for an and-node
to be satisfied, all branches must be satisfied.
In the tree, the characteristic reward related to election poses the question
whether or not the reward can only be handed out by the election winner. An
example of such rewards is granting amnesty, which is not possible unless elected
As not all aspects of vote influencing are of direct interest to the buyer (e.g.
how sure is delivery of the reward), other trees for different objectives have been
constructed. Noting that this objective tree focuses upon vote acquisition only
via rewarding leads to the following dimension:
type of compliance how is compliance achieved?
instrumental (by rewarding), normative (by convincing), coercive.
A vote buyer, however,is not interested in acquiring one vote, but in acquiring
many votes. Specifically, a vote buyer’s goal is to acquire enough votes to make a
difference. This goal of a vote buyer encompasses the above objective of acquiring
one vote. A vote buyer could use various means to attempt vote acquisition of
many votes (as indicated by the examples). An analysis of this, similar to the
above one, uncovered the following dimensions of approaches to vote buying:
non-compliance what impact does non-compliance have?
no rewarding, rewarded anyway.
focusability can only the relevant people be the sole targets of the vote buying
highly targetable, less targetable.
scalability how easy is it to employ this rewarding on a large scale?
high, medium, low.
costs do costs vary with the number of acquired votes or voters convinced?
variable, more fixed.
openness/publicity is the persuasion attempt general knowledge?
secret (known to buyer and seller(s)), not hidden, general knowledge.
Considering the point of view of the vote seller introduced the following
dimensions for vote buying:
rewarding certainty can rewarding be avoided?
consequences of non-reward what impact will not rewarding have?
high impact, low impact.
Note that these two dimensions are closely related; they can be combined as
commitment to reward.
Considering these various objectives together (i.e. vote seller, vote buyer for one
vote and for a group of votes) led to the following dimension:
proof of compliance who should prove compliance?
proof by buyer, proof by seller. in case of vote buying, proof by seller is
expected; in case of promises, proof by buyer is expected
One remarkable observation, given these dimensions, is that absence of re-
ceipts (receipt-freeness, see e.g. ) is not sufficient to prevent vote buying –
it only suffices to prevent proving compliance. Forms of vote buying exist that
do not require proving compliance by the individual voter, as can be seen in
2.3Classifying the targets
The set of possible targets for voter influencing can be characterised on various
levels. From large to small, we distinguish the following:
⊇ eligible voters
⊇ registered voters
⊇ voters casting votes
⊇ voters casting valid votes
⊇ voters casting valid votes for vote buyer(6)
Additionally, preferences with respect to elections and vote buying differ from
person to person. Perhaps some individuals do not mind selling their votes, while
others may find the practice so repugnant they will not vote for anyone involved
with the practice – even if it is their preferred candidate. We distinguish the
following dimensions for voters:
will accept reward
awareness of attempt none / heard rumours / fully aware
is a desired targetyes / no
cast vote buyer’s choice / other
yes / no
buyer’s choice / other
This classification can be applied to each of the sets 1–6. This classification
extends the work of Acker , who classified voters targeted by election promises
A. Already compliant voters — these would have voted for the coercer without
the election promise
B. Voters who change their votes — these vote for the coercer due to the election
C. Non-compliant voters — these do not vote for the coercer, despite the election
One category missing in that classification is explicitly included by our new
classification: the set of voters who, as a result of the vote buying attempt,
change their vote from “buyer’s choice” to “other”. Intuitively, these voters can
be characterised as the voters who find vote buying so repugnant, that they will
not vote for anyone involved with the practice.
2.4Classification of the examples
Below we classify our examples of voter influencing, according to the dimensions2
established above. In addition, for each example we note which subset of voters
Example 1 (handout). conditions: unconditional; group size: individual; re-
lated: no; type: instrumental; non-compliance: rewarded; focus: not tar-
getable; scalable: low; costs: number of attempts; publicity: not hidden; com-
mitment: unavoidable, low; proof: none.
Targetted at voters casting votes (class 4).
Example 2 (theme park). conditions: conditional; group size: collective; re-
lated: yes; type: instrumental; non-compliance: rewarded; focus: not tar-
getable; scalable: high; costs: fixed; publicity: public; commitment: avoid-
able, high; proof: none.
Targetted at registered voters (class 3).
Example 3 (zalmsnip). conditions: conditional; group size: collective; related:
yes; type: instrumental; non-compliance: rewarded; focus: not targetable;
scalable: high; costs: fixed; publicity: public; commitment: avoidable, high;
Targetted at registered voters (class 3).
Example 4 (election promise). conditions: conditional; group size: collective;
related: yes; type: instrumental; non-compliance: rewarded; focus: highly
targetable; scalable: high; costs: fixed; publicity: public; commitment: avoid-
able, high; proof: none.
Targetted at a subgroup of registered voters (class 3).
Example 5 (rf). conditions: conditional; group size: individual; related: no;
type: instrumental; non-compliance: unrewarded; focus: highly targetable;
scalable: low; costs: number of acquired vote; publicity: not hidden; com-
mitment: avoidable, low; proof: by voter.
Targetted at voters casting valid votes (class 5).
Example 6 (non-rf). conditions: conditional; group size: individual; related:
no; type: instrumental; non-compliance: unrewarded; focus: highly targetable;
scalable: low; costs: number of compliant voters; publicity: not hidden; com-
mitment: avoidable, low; proof: by seller.
Targetted at voters casting valid votes (class 5).
2Both time of reward and type of reward have been left out, as these are already
explicit in the examples.
Example 7 (reimburse). conditions: unconditional; group size: individual; re-
lated: no; type: instrumental; non-compliance: rewarded; focus: not tar-
getable; scalable: medium; costs: number of requesting voters; publicity: pub-
lic; commitment: unavoidable, –; proof: not required.
Targetted at registered voters (class 3).
We find that, based on our distinctions, we can easily classify these examples.
The question remains which attributes indicate acceptable and unacceptable
forms, respectively. From our examples and their intuitive acceptability, we pro-
pose that benefits that are related to the contested position, are unconditional
and openly announced, are most likely to be found legitimate.
2.5 Conclusions on vote influencing
We conclude that vote buying involves much more than offering money in ex-
change for a proof of compliance. Attributes that make it likely for an action to
be considered vote buying include:
– unrelated to contested position;
– reward independent of being elected;
– reward conditional on compliance (therefore proof by seller);
– highly targetable;
– variable costs (related to individual payment);
– secrecy of the attempt.
Individuality does not make things worse; buying a whole district is in itself
no better than buying votes one by one. The publication of any election result on
a level lower than strictly necessary (e.g. per polling station) facilitates collective
vote buying, and would best be eliminated from this perspective. Electronic
voting can facilitate such a transition, by storing votes independently from the
place where they were cast.
However, this is by itself not sufficient to stop collective vote buying. If a
buyer wants to buy a set of votes, and knows with 70% certainty that an indi-
vidual voter complies, she can be fairly sure that if she buys a large amount of
votes, 70% of the votes will be hers, due to the law of large numbers.
Conversely, if in a particular set of votes 70% is for the buyer, she can derive
that a voter whose vote is in this set has voted for her with a 70% probability.
This particular observation gives rise to the notion of probabilistic vote buying –
where a buyer requires not exact votes, but is satisfied by a (significant) change
in the distribution of votes.
Furthermore, the non-rf example showed that it is possible to discourage
voting for a certain party or candidate in an action of vote buyinng. This is an
example of voter influencing that cannot be directly described in terms of the
intuitive notion of receipt-freeness, but is close to it.
In the next section, we categorise anonymity in voting based on these obser-
3Dimensions of anonymity in voting
Traditionally, the concept of vote buying has been related to the possibility of
providing a proof of one’s choice. The notion of receipt-freeness was proposed
to prevent such a proof. However, our framework developed in the previous
sections shows that a proof is not always necessary. The following actions would
be possible without a proof of the voter’s choice in a strict sense:
– rewarding the voter if she does not vote for a specific party or candidate
(related to negative proof);
– rewarding the voter if it is likely that she made a certain choice.
It can be enough for a buyer to hand out the reward if a voter can show
that she did not vote for two of the buyer’s opponents (example 6). The buyer
could also pay a voter if after observing the outcome, it is more likely that this
voter voted for him than that another voter voted for him. If this can be observed
from messages sent in the voting protocol, this should be addressed by computer
science verification methods. One could also derive this from voter behaviour ,
but that is hard to prevent using computer science tools.
Each of the notions of privacy, receipt-freeness and coercion-resistance can
be investigated with respect to these scenarios:
Barghava et al.
coercion-resistance Delaune et al.
The distinction between the three notions of anonymity depends on the rela-
tion between voter and attacker. In case of privacy, no cooperation of the voter
is assumed; the attacker tries to find out about the voter’s choice without co-
operation the voter. In case of receipt-freeness, the voter cooperates by sending
information to the attacker. In case of coercion-resistance, the voter even accepts
instructions from the attacker.
The weak variant of the notion then concerns the situation that an attacker
cannot be sure of the voter’s choice. The strong variant of the notion concerns
the situation that an attacker cannot be sure of what the voter did not choose
either. The probabilistic variant of the notion concerns the situation what an
attacker cannot deduce anything from the probability distribution of a voter’s
In the next section, we formalise the notions of strong and weak privacy
and receipt-freeness in an epistemic framework. Probabilistic privacy has been
investigated in  (where it is called probabilistic anonymity). The notion of
coercion-resistance is defined in . Filling the remaining fields in the table
based on these definitions is future work.
4 Formalising anonymity properties
Garcia, Hasuo, Pieters and Van Rossum defined a framework for describing
anonymity properties of protocols in epistemic logic . We use this framework
as the basis for our definitions of privacy and receipt-freeness. Their definitions
are based on a formalisation of runs in protocols. For the formal definitions, we
refer to the original paper.
Based on observational equivalence of runs, the following notions are defined
formally in the original paper. We include the informal definitions here. The
formula A Sends m to B means: at some stage in the run, A sends a message to
B which contains m as a subterm. A Sends m means that A sends the message
m to someone. The formula A Possesses m means: after the run has finished,
A is capable of constructing the message m. The formula A Originates m means
that A Sends m, but A is not relaying. More precisely, m does not appear as a
subterm of a message which A has received before.
The formula ?Aϕ is read as “after the run is completed, the agent A knows
that ϕ is true”. The formula ♦Aϕ is short for ¬?A¬ϕ and read as “after the
run is completed, the agent A suspects that ϕ is true”.
Garcia et al. define the information hiding properties of sender anonymity,
unlinkability and plausible deniability using the notion of an anonymity set (a
collection of agents among which a given agent is not identifiable) in epistemic
logic as follows:
Definition 1. (Sender anonymity) Suppose that r is a run of a protocol in which
an agent B receives a message m. We say that r provides sender anonymity with
anonymity set AS if it satisfies
♦B(X Originates m).
This means that, as far as B is concerned, every agent in the anonymity set
could have sent the message.
Definition 2. (Unlinkability) A run r provides unlinkability for users A and B
with anonymity set AS iff
where ϕ(X,Y ) = ∃n.?X Sends n ∧ Y Possesses n?.
Intuitively, the left side of the conjunction means that the spy (the adversary)
is not certain that A sent something to B. The right conjunct means that,
according to the spy, every other user could have sent a message to B. Sim-
ilarly, unlinkability between a user A and a message m could be defined as
|= ¬?spy(A Sends m) ∧?
In certain circumstances (e.g. relays), agents might be interested in showing
that they did not know that they had some sensitive information m. This might
be modeled by the following epistemic formula:
X∈AS♦spy(X Sends m).
Definition 3. (Plausible deniability) Agent A can plausibly deny message m in
run r iff
r |= ?spy¬(?A(A Possesses m)) .
This formula is read as: the spy knows that A does not know that she possesses
We extend this set of definitions by providing the additional property of receipt-
freeness. Receipt-freeness of an agent A with respect to a message m (e.g. a vote)
intuitively means that A cannot send a message m′to the spy that proves that
she sent m in the past. For this purpose, the definition of plausible deniability
is too strong, since A does know that she possesses m. Sender anonymity is par-
ticularly useful for providing anonymity of the voter with respect to the election
authorities, but in receipt-freeness, A herself tries to communicate with the spy.
Instead, it should not be possible to link A to her vote. Thus, unlinkability seems
the most natural property to base our definition of receipt-freeness upon.
In the anonymity framework, the concept of anonymity set is used to define
the set of entities between which an observer should not be able to distinguish.
To apply the framework to votes, we need to adapt the concept of anonymity
set. In voting, we are sure that each (actual) voter submits a vote. Therefore,
the point is not whether any other user in an anonymity set could have sent the
message, but whether the voter could have submitted any other vote. Therefore,
we define an anonymity set of messages, AMS, instead of an anonymity set of
agents. This set typically consists of all possible votes.
First of all, we define the notion of (weak) privacy. This can be achieved
without referring to the anonymity set.
Definition 4. (Weak privacy) A run of a protocol is weakly private for agent
A with respect to message m iff
r |= ¬?spy(A Sends m)
To be able to define receipt-freeness, we need to have a way to extend a
given run with one message: the receipt. We write this as r.(A → B : m)) for
a given run r, message m (the receipt), sender A and receiver B. For A to be
able to send the receipt, she needs to have the message in her possessions at the
end of the original run. The new run does not need to be a run of the protocol.
It does need to be legitimate with respect to the initial possession function.
PossIPo(r,A,|r|−1) denotes the possessions of agent A at the end of the original
run (see ).
Definition 5. (Weak receipt-freeness) A run of a protocol is weakly receipt-free
for agent A with respect to message m iff for all m′∈ PossIPo(r,A,|r| − 1),
r.(A → spy : m′) |= ¬?spy(A Sends m)
Weak receipt-freeness implies that the voter cannot prove to the spy that she sent
message m during the protocol, where m is the (part of a) message representing
the vote. However, this notion is still fairly limited. For example, suppose that
the spy wants the voter to vote for party X. Suppose, furthermore, that the
voter instead chooses to vote Y , which is represented by message m in the above
definition. Now, if the voter cannot show that she voted Y , this protocol is
receipt-free with respect to the definition above. However, if the spy can acquire
information which proves that the voter did not vote X, the spy will not be
satisfied. Therefore, we introduce stronger notions of privacy and receipt-freeness
Definition 6. (Strong privacy) A run of a protocol is strongly private for agent
A with respect to a message m in anonymity set AMS iff
r |= (¬?spy(A Sends m)) ∧
♦spy(A Sends m′′)
Definition 7. (Strong receipt-freeness) A run of a protocol is strongly receipt-
free for agent A with respect to a message m in anonymity set AMS iff for all
m′∈ PossIPo(r,A,|r| − 1),
r.(A → spy : m′) |= (¬?spy(A Sends m)) ∧
♦spy(A Sends m′′)
Here, no matter what information the voter supplies to the spy, any vote in
the anonymity set is still possible. This is represented by the “suspects” symbol
♦spy. In other words, for all possible votes, the spy still suspects that the voter
cast this particular vote; or: the spy is not certain she did not cast this vote.
This requires that at least one message has been received (i.e. at least one vote
has been cast) for every message (vote) m′′∈ AMS. Otherwise, the spy could
observe from the results that no-one, in particular not voter A, cast a certain
vote. Thus, for votes, AMS is a subset of the set of candidates who received
Notice that this definition is analogous to the definition of unlinkability of
Garcia et al.
Theorem 1. If a run of a protocol is strongly receipt-free for agent A with
respect to message m in anonymity set AMS, then it is also weakly receipt-free
for agent A with respect to message m.
Proof. This follows directly from the definitions.
In the framework, it can be defined which part of the messages the spy can
observe. If the spy could read all the messages, the voter only needs to supply
the secret keys in order to provide a receipt. This is not what is commonly
understood by analyzing receipt-freeness. Instead, there are certain messages in
the protocol that the spy is not assumed to have access to (when the voter is in
a “voting booth”).
In our definition, we deviate from the approach by Delaune et al. . In-
tuitively, receipt-freeness is achieved if a voter does not possess convincing,
exclusive evidence of how she voted. The approach by Delaune et al. defines
receipt-freeness using two voters (to preserve indistinguishability of the result).
By focusing on the actual receipt, our definition only relies on one voter, and
thus remains closer to the intuition. The indistinguishability is made explicit in
our definition by AMS, which does not need to encompass all candidates but can
be confined e.g. to all candidates for whom at least one vote was cast.
5 Conclusions and future work
In this paper, we examined various dimensions of the notion of anonymity in
electronic voting. Based on an analysis of the acceptability of vote buying and
coercion, we found that many different dimensions contribute to whether an
action is classified as vote buying or coercion, or not. Having established these
dimensions, we distinguished between weak, strong and probabilistic notions of
anonymity in voting. This distinction applies to privacy, receipt-freeness and
Following the definitions of anonymity in , we introduced an approach to
formally verify weak and strong receipt-freeness in epistemic logic. To the best
of our knowledge, we are the first to do so. The approach has not been tested
on real protocols thus far.
One of the main benefits of our approach is the intuitive definition that it
provides for receipt-freeness. As opposed to other approaches, especially , the
“receipt” can easily be distinguished in our model as a separate message that
the voter sends to the spy. Instead of investigating whether the spy can recover
the vote from forwarded messages, we judge whether the spy really knows what
the voter’s choice was, based on any possible receipt. This notion of knows is
characteristic for the epistemic logic approach, and this justifies our choice for
the anonymity framework of  as a basis.
While our approach captures receipt-freeness, the investigation of the notion
of vote influencing clearly indicates that compliance with a technical criterion
such as receipt-freeness is insufficient to prevent voter influencing – if unaware,
a voter could still be susceptible to voter influencing. Further study in this area
(see also ) is needed to ensure that security requirements not only enable
secure voting, but that voters are sufficiently aware of the security they provide.
In future work, we aim at providing an alternative definition of receipt-
freeness in our model, based on the knowledge of the spy instead of on extension
of a run. We hope to prove that the two definitions are equivalent. Moreover,
we wish to apply the definitions to existing voting protocols, in order to prove
(or disprove) receipt-freeness. It may also be interesting to investigate the rela-
tion between verifiability  and receipt-freeness in epistemic logic, since both
receipt-freeness and verifiability are based on an agent’s knowledge instead of its
As we have seen, voter influencing can take many forms, and only some can
be hindered by technological means. However, being more precise in what we can
and cannot achieve in technology can lead to better decisions on which protocols
are acceptable and which are not. Again, the latter will depend on what is seen
as acceptable in a particular culture, which has been determined during a long
history of success and fraud.
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