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Homomorphic signature schemes allow anyone to perform computation on signed data in such a way that the correctness of computation’s results is publicly certified. In this work we analyze the security notions for this powerful primitive considered in previous work, with a special focus on adaptive security. Motivated by the complications of existing security models in the adaptive setting, we consider a simpler and (at the same time) stronger security definition inspired to that proposed by Gennaro and Wichs (ASIACRYPT’13) for homomorphic MACs. In addition to strength and simplicity, this definition has the advantage to enable the adoption of homomorphic signatures in dynamic data outsourcing scenarios, such as delegation of computation on data streams. Then, since no existing homomorphic signature satisfies this stronger notion, our main technical contribution are general compilers which turn a homomorphic signature scheme secure under a weak definition into one secure under the new stronger notion. Our compilers are totally generic with respect to the underlying scheme. Moreover, they preserve three important properties of homomorphic signatures: composability, context-hiding (i.e. signatures on computation’s output do not reveal information about the input) and efficient verification (i.e. verifying a signature against a program \({\mathcal P}\) can be made faster, in an amortized, asymptotic sense, than recomputing \({\mathcal P}\) from scratch).

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... However, this would prevent the user from remaining in control of her data and require the data source to bear computations that are outside of its interests. Another approach is to let the data source certify the user's data by using specialized digital signature schemes such as homomorphic signatures [21,26,27,28,57] or homomorphic authenticators [5,49,55,88]. Thus, the user could locally compute the queried operation and provide the service provider with the result and a homomorphic signature attesting its correct computation on her data. ...

... Although VE can be used for proofs of correct encryption, it does not address data authenticity, which is ensured by cryptographic techniques, such as hash functions, that are more complex than polynomial relations. To this end, homomorphic signatures [21,26,28,57] and homomorphic authenticators [5,49,55,88] enable privacy-preserving computations on authenticated data. In particular, such schemes produce a signature of the plaintext result of homomorphic computations without deciphering. ...

In the digital era, users share their personal data with service providers to obtain some utility, e.g., access to high-quality services. Yet, the induced information flows raise privacy and integrity concerns. Consequently, cautious users may want to protect their privacy by minimizing the amount of information they disclose to curious service providers. Service providers are interested in verifying the integrity of the users' data to improve their services and obtain useful knowledge for their business. In this work, we present a generic solution to the trade-off between privacy, integrity, and utility, by achieving authenticity verification of data that has been encrypted for offloading to service providers. Based on lattice-based homomorphic encryption and commitments, as well as zero-knowledge proofs, our construction enables a service provider to process and reuse third-party signed data in a privacy-friendly manner with integrity guarantees. We evaluate our solution on different use cases such as smart-metering, disease susceptibility, and location-based activity tracking, thus showing its promising applications. Our solution achieves broad generality, quantum-resistance, and relaxes some assumptions of state-of-the-art solutions without affecting performance.

... We might use a hash tree for this purpose, as in [3] and [5], but that technique is not compatible with multi-dataset. Instead, we will adapt a technique using an OR-homomorphic authenticator given by Catalano, Fiore, and Nizzardo [10]. ...

Homomorphic authenticated encryption allows implicit computation on plaintexts using corresponding ciphertexts without losing privacy, and provides authenticity of the computation and the resultant plaintext of the computation when performing a decryption. However, due to its special functionality, the security notions of the homomorphic authenticated encryption is somewhat complicated and the construction of fully homomorphic authenticated encryption has never been given. In this work, we propose a new security notion and the first construction of fully homomorphic authenticated encryption. Our new security notion is a unified definition for data privacy and authenticity of homomorphic authenticated encryption. Moreover, our security notion is simpler and stronger than the previous ones. To realize our new security notion, we also suggest a construction of fully homomorphic authenticated encryption via generic construction. We combine a fully homomorphic encryption and two homomorphic authenticators, one fully homomorphic and one OR-homomorphic, to construct a fully homomorphic authenticated encryption that satisfies our security notion. Our construction requires its fully homomorphic encryption to be indistinguishable under chosen plaintext attacks and its homomorphic authenticators to be unforgeable under selectively chosen plaintext queries. Our construction also supports multiple datasets and amortized efficiency. For efficiency, we also construct a multi-dataset fully homomorphic authenticator scheme, which is a variant of the first fully homomorphic signature scheme. Our multi-dataset fully homomorphic authenticator scheme satisfies the security requirement of our generic construction above and supports amortized efficiency.

Functional Commitments (FC) allow one to reveal functions of committed data in a succinct and verifiable way. In this paper we put forward the notion of additive-homomorphic FC and show two efficient, pairing-based, realizations of this primitive supporting multivariate polynomials of constant degree and monotone span programs, respectively. We also show applications of the new primitive in the contexts of homomorphic signatures: we show that additive-homomorphic FCs can be used to realize homomorphic signatures (supporting the same class of functionalities as the underlying FC) in a simple and elegant way. Using our new FCs as underlying building blocks, this leads to the (seemingly) first expressive realizations of multi-input homomorphic signatures not relying on lattices or multilinear maps.

Data authentication primarily serves as a tool to achieve data integrity and source authentication. However, traditional data authentication does not fit well where an intermediate entity (editor) is required to modify the authenticated data provided by the source/data owner before sending the data to other recipients. To ask the data owner for authenticating each modified data can lead to higher communication overhead. In this work, we introduce the notion of editing-enabled signatures where the data owner can choose any set of modification operations applicable on the data and still can restrict any possibly untrusted editor to authenticate the data modified using an operation from this set only. Moreover, the editor does not need to interact with the data owner in order to authenticate the data every time it is modified. We construct an editing-enabled signature scheme that derives its efficiency from mostly lightweight cryptographic primitives. We formalize the security model for editing-enabled signatures and analyze the security of our editing-enabled signature scheme. Editing-enabled signatures can find numerous applications that involve generic editing tasks and privacy-preserving operations. We demonstrate how our editing-enabled signature scheme can be applied in two privacy-preserving applications.

We introduce the notion of asymmetric programmable hash functions (APHFs, for short), which adapts Programmable hash functions, introduced by Hofheinz and Kiltz (Crypto 2008, Springer, 2008), with two main differences. First, an APHF works over bilinear groups, and it is asymmetric in the sense that, while only secretly computable, it admits an isomorphic copy which is publicly computable. Second, in addition to the usual programmability, APHFs may have an alternative property that we call programmable pseudorandomness. In a nutshell, this property states that it is possible to embed a pseudorandom value as part of the function’s output, akin to a random oracle. In spite of the apparent limitation of being only secretly computable, APHFs turn out to be surprisingly powerful objects. We show that they can be used to generically implement both regular and linearly-homomorphic signature schemes in a simple and elegant way. More importantly, when instantiating these generic constructions with our concrete realizations of APHFs, we obtain: (1) the first linearly-homomorphic signature (in the standard model) whose public key is sub-linear in both the dataset size and the dimension of the signed vectors; (2) short signatures (in the standard model) whose public key is shorter than those by Hofheinz–Jager–Kiltz (Asiacrypt 2011, Springer, 2011) and essentially the same as those by Yamada et al. (CT-RSA 2012, Springer, 2012).

In this paper we explore a powerful extension of the notion of pseudo-free groups, proposed by Rivest at TCC 2004. We identify,
motivate, and study pseudo-freeness in face of adaptive adversaries who may learn solutions to other non-trivial equations before having to solve a new non-trivial equation.
We present a novel, carefully crafted definition of adaptive pseudo-freeness that walks a fine line between being too weak and being unsatisfiable. We show that groups that satisfy our
definition yield, via a generic construction, digital and network coding signature schemes.
Finally, we obtain concrete constructions of such schemes in the RSA group by showing this group to be adaptive pseudo-free.
In particular, we demonstrate the generality of our framework for signatures by showing that most existing schemes are instantiations
of our generic construction.

In tandem with recent progress on computing on encrypted data via fully homomorphic encryption, we present a framework for computing on authenticated data via the notion of slightly homomorphic signatures, or P-homomorphic signatures. With such signatures, it is possible for a third party to derive a signature on the object m′ from a signature of m as long as P(m,m′) = 1 for some predicate P which captures the “authenticatable relationship” between m′ and m. Moreover, a derived signature on m′ reveals no extra information about the parent m.
Our definition is carefully formulated to provide one unified framework for a variety of distinct concepts in this area, including arithmetic, homomorphic, quotable, redactable, transitive signatures and more. It includes being unable to distinguish a derived signature from a fresh one even when given the original signature. The inability to link derived signatures to their original sources prevents some practical privacy and linking attacks, which is a challenge not satisfied by most prior works.
Under this strong definition, we then provide generic constructions for all univariate and closed predicates, and specific efficient constructions for a broad class of natural predicates such as quoting, subsets, weighted sums, averages, and Fourier transforms. To our knowledge, these are the first efficient constructions for these predicates (excluding subsets) that provably satisfy this strong security notion.

In a homomorphic signature scheme, a user Alice signs some large dataset x using her secret signing key and uploads the signed data to an untrusted remote server. The server can then run some computation y=f(x) over the signed data and homomorphically derive a short signature σ[subscript f,y] certifying that y is the correct output of the computation f. Anybody can verify the tuple (f, y, σ[subscript f,y]) using Alice's public verification key and become convinced of this fact without having to retrieve the entire underlying data. In this work, we construct the first leveled fully homomorphic signature} schemes that can evaluate arbitrary {circuits} over signed data. Only the maximal {depth} d of the circuits needs to be fixed a-priori at setup, and the size of the evaluated signature grows polynomially in d, but is otherwise independent of the circuit size or the data size. Our solution is based on the (sub-exponential) hardness of the small integer solution (SIS) problem in standard lattices and satisfies full (adaptive) security. In the standard model, we get a scheme with large public parameters whose size exceeds the total size of a dataset. In the random-oracle model, we get a scheme with short public parameters. In both cases, the schemes can be used to sign many different datasets. The complexity of verifying a signature for a computation f is at least as large as that of computing f, but can be amortized when verifying the same computation over many different datasets. Furthermore, the signatures can be made context-hiding so as not to reveal anything about the data beyond the outcome of the computation. These results offer a significant improvement in capabilities and assumptions over the best prior homomorphic signature schemes, which were limited to evaluating polynomials of constant degree. As a building block of independent interest, we introduce a new notion called homomorphic trapdoor functions (HTDF) which conceptually unites homomorphic encryption and signatures. We construct HTDFs by relying on the techniques developed by Gentry et al. (CRYPTO '13) and Boneh et al. (EUROCRYPT '14) in the contexts of fully homomorphic and attribute-based encryptions.

We introduce the notion of asymmetric programmable hash functions (APHFs, for short), which adapts Programmable Hash Functions, introduced by Hofheinz and Kiltz at Crypto 2008, with two main differences. First, an APHF works over bilinear groups, and it is asymmetric in the sense that, while only secretly computable, it admits an isomorphic copy which is publicly computable. Second, in addition to the usual programmability, APHFs may have an alternative property that we call programmable pseudorandomness. In a nutshell, this property states that it is possible to embed a pseudorandom value as part of the function’s output, akin to a random oracle. In spite of the apparent limitation of being only secretly computable, APHFs turn out to be surprisingly powerful objects. We show that they can be used to generically implement both regular and linearly-homomorphic signature schemes in a simple and elegant way. More importantly, when instantiating these generic constructions with our concrete realizations of APHFs, we obtain: (1) the first linearly-homomorphic signature (in the standard model) whose public key is sub-linear in both the dataset size and the dimension of the signed vectors; (2) short signatures (in the standard model) whose public key is shorter than those by Hofheinz-Jager-Kiltz from Asiacrypt 2011, and essentially the same as those by Yamada, Hannoka, Kunihiro, (CT-RSA 2012).

In a homomorphic signature scheme, a user Alice signs some large dataset x using her secret signing key and uploads the signed data to an untrusted remote server. The server can then run some computation y=f(x) over the signed data and homomorphically derive a short signature σf,y certifying that y is the correct output of the computation f. Anybody can verify the tuple (f, y, σf,y) using Alice's public verification key and become convinced of this fact without having to retrieve the entire underlying data. In this work, we construct the first leveled fully homomorphic signature} schemes that can evaluate arbitrary {circuits} over signed data. Only the maximal {depth} d of the circuits needs to be fixed a-priori at setup, and the size of the evaluated signature grows polynomially in d, but is otherwise independent of the circuit size or the data size. Our solution is based on the (sub-exponential) hardness of the small integer solution (SIS) problem in standard lattices and satisfies full (adaptive) security. In the standard model, we get a scheme with large public parameters whose size exceeds the total size of a dataset. In the random-oracle model, we get a scheme with short public parameters. In both cases, the schemes can be used to sign many different datasets. The complexity of verifying a signature for a computation f is at least as large as that of computing f, but can be amortized when verifying the same computation over many different datasets. Furthermore, the signatures can be made context-hiding so as not to reveal anything about the data beyond the outcome of the computation.
These results offer a significant improvement in capabilities and assumptions over the best prior homomorphic signature schemes, which were limited to evaluating polynomials of constant degree.
As a building block of independent interest, we introduce a new notion called homomorphic trapdoor functions (HTDF) which conceptually unites homomorphic encryption and signatures. We construct HTDFs by relying on the techniques developed by Gentry et al. (CRYPTO '13) and Boneh et al. (EUROCRYPT '14) in the contexts of fully homomorphic and attribute-based encryptions.

A signature scheme is malleable if, on input a message and a signature, it is possible to efficiently compute a signature on a related message, for a transformation that is allowed with respect to this signature scheme. In this paper, we first provide new definitions for malleable signatures that allow us to capture a broader range of transformations than was previously possible. We then give a generic construction based on malleable zero-knowledge proofs that allows us to construct malleable signatures for a wide range of transformation classes, with security properties that are stronger than those that have been achieved previously. Finally, we construct delegatable anonymous credentials from signatures that are malleable with respect to an appropriate class of transformations (that we show our malleable signature supports). The resulting instantiation satisfies a stronger security notion than previous schemes while also scaling linearly with the number of delegations.

Homomorphic MACs, introduced by Gennaro and Wichs in 2013, allow anyone to validate computations on authenticated data without knowledge of the secret key.Moreover, the secret-key owner can verify the validity of the computation without needing to know the original (authenticated) inputs. Beyond security, homomorphic MACs are required to produce short tags (succinctness) and to support composability (i.e., outputs of authenticated computations should be re-usable as inputs for new computations).
At Eurocrypt 2013, Catalano and Fiore proposed two realizations of homomorphic MACs that support a restricted class of computations (arithmetic circuits of polynomial degree), are practically efficient, but fail to achieve both succinctness and composability at the same time.
In this paper, we generalize the work of Catalano and Fiore in several ways. First, we abstract away their results using the notion of encodings with limited malleability, thus yielding new schemes based on different algebraic settings. Next, we generalize their constructions to work with graded encodings, and more abstractly with k-linear groups. The main advantage of this latter approach is that it allows for homomorphic MACs which are (somewhat) composable while retaining succinctness. Interestingly, our construction uses graded encodings in a generic way. Thus, all its limitations (limited composability and non-constant size of the tags) solely depend on the fact that currently known multilinear maps share similar constraints. This means, for instance, that our scheme would support arbitrary circuits (polynomial depth) if we had compact multilinear maps with an exponential number of levels.

Homomorphic signatures are primitives that allow for public computations for a class of specified predicates over authenticated data. An enhanced privacy notion, called complete context-hiding security, was recently motivated by Attrapadung et al. (Asiacrypt’12). This notion ensures that a signature derived from any valid signatures is perfectly indistinguishable from a newly generated signatures (on the same message), and seems desirable in many applications requiring to compute on authenticated data. In this paper, we focus on two useful predicates – namely, substring quotation predicates and linear dependency predicates – and present the first completely context-hiding schemes for these in the standard model. Moreover, our new quotable signature scheme is the first such construction with signatures of linear size. In comparison with the initial scheme of Ahn et al. (TCC 2012), we thus reduce the signature size from O(nlogn) to O(n), where n is the message size. Our scheme also allows signing messages of arbitrary length using constant-size public keys.

We define and construct a new primitive called a fully homomorphic message authenticator. With such scheme, anybody can perform arbitrary computations over authenticated data and produce a short tag that authenticates the result of the computation (without knowing the secret key). This tag can be verified using the secret key to ensure that the claimed result is indeed the correct output of the specified computation over previously authenticated data (without knowing the underlying data). For example, Alice can upload authenticated data to “the cloud”, which then performs some specified computations over this data and sends the output to Bob, along with a short tag that convinces Bob of correctness. Alice and Bob only share a secret key, and Bob never needs to know Alice’s underlying data. Our construction relies on fully homomorphic encryption to build fully homomorphic message authenticators.

A homomorphic signature scheme for a class of functions \(\mathcal{C}\) allows a client to sign and upload elements of some data set D on a server. At any later point, the server can derive a (publicly verifiable) signature that certifies that some y is the result computing some \(f\in\mathcal{C}\) on the basic data set D. This primitive has been formalized by Boneh and Freeman (Eurocrypt 2011) who also proposed the only known construction for the class of multivariate polynomials of fixed degree d ≥ 1. In this paper we construct new homomorphic signature schemes for such functions. Our schemes provide the first alternatives to the one of Boneh-Freeman, and improve over their solution in three main aspects. First, our schemes do not rely on random oracles. Second, we obtain security in a stronger fully-adaptive model: while the solution of Boneh-Freeman requires the adversary to query messages in a given data set all at once, our schemes can tolerate adversaries that query one message at a time, in a fully-adaptive way. Third, signature verification is more efficient (in an amortized sense) than computing the function from scratch. The latter property opens the way to using homomorphic signatures for publicly-verifiable computation on outsourced data. Our schemes rely on a new assumption on leveled graded encodings which we show to hold in a generic model.

Homomorphic message authenticators allow the holder of a (public) evaluation key to perform computations over previously authenticated data, in such a way that the produced tag σ can be used to certify the authenticity of the computation. More precisely, a user knowing the secret key sk used to authenticate the original data, can verify that σ authenticates the correct output of the computation. This primitive has been recently formalized by Gennaro and Wichs, who also showed how to realize it from fully homomorphic encryption. In this paper, we show new constructions of this primitive that, while supporting a smaller set of functionalities (i.e., polynomially-bounded arithmetic circuits as opposite to boolean ones), are much more efficient and easy to implement. Moreover, our schemes can tolerate any number of (malicious) verification queries. Our first construction relies on the sole assumption that one way functions exist, allows for arbitrary composition (i.e., outputs of previously authenticated computations can be used as inputs for new ones) but has the drawback that the size of the produced tags grows with the degree of the circuit. Our second solution, relying on the D-Diffie-Hellman inversion assumption, offers somewhat orthogonal features as it allows for very short tags (one single group element!) but poses some restrictions on the composition side.

Homomorphic signatures are primitives that allow for public computations on authenticated data. At TCC 2012, Ahn et al. defined a framework and security notions for such systems. For a predicate P, their notion of P-homomorphic signature makes it possible, given signatures on a message set M, to publicly derive a signature on any message m′ such that P(M,m′)=1. Beyond unforgeability, Ahn et al. considered a strong notion of privacy --- called strong context hiding --- requiring that derived signatures be perfectly indistinguishable from signatures newly generated by the signer. In this paper, we first note that the definition of strong context hiding may not imply unlinkability properties that can be expected from homomorphic signatures in certain situations. We then suggest other definitions of privacy and discuss the relations among them. Our strongest definition, called complete context hiding security, is shown to imply previous ones. In the case of linearly homomorphic signatures, we only attain a slightly weaker level of privacy which is nevertheless stronger than in previous realizations in the standard model. For subset predicates, we prove that our strongest notion of privacy is satisfiable and describe a completely context hiding system with constant-size public keys. In the standard model, this construction is the first one that allows signing messages of arbitrary length. The scheme builds on techniques that are very different from those of Ahn et al.

In this paper we introduce the notion of Algebraic (Trapdoor) One Way Functions, which, roughly speaking, captures and formalizes many of the properties of number-theoretic one-way functions. Informally, a (trapdoor) one way function F: X → Y is said to be algebraic if X and Y are (finite) abelian cyclic groups, the function is homomorphic i.e. F(x)·F(y) = F(x ·y), and is ring-homomorphic, meaning that it is possible to compute linear operations “in the exponent” over some ring (which may be different from ℤp
where p is the order of the underlying group X) without knowing the bases. Moreover, algebraic OWFs must be flexibly one-way in the sense that given y = F(x), it must be infeasible to compute (x′, d) such that F(x′) = y
d
(for d ≠ 0). Interestingly, algebraic one way functions can be constructed from a variety of standard number theoretic assumptions, such as RSA, Factoring and CDH over bilinear groups.
As a second contribution of this paper, we show several applications where algebraic (trapdoor) OWFs turn out to be useful. These include publicly verifiable secure outsourcing of polynomials, linearly homomorphic signatures and batch execution of Sigma protocols.

The security of modern networked computers is very low and must be dramatically improved. Integrity of data and programs is an essential aspect of computers. We propose approaches towards computer security in which the main trust is a cryptographically authenticated “keyboard”. The achievability follows from the current trend towards personal computers, workstations and notebooks. We discuss how this could increase computer security and which problems remain to be solved in such an environment. 1

We propose a general framework that converts (ordinary) signature schemes having certain properties into linearly homomorphic signature schemes, i.e., schemes that allow authentication of linear functions on signed data. The security of the homomorphic scheme follows from the same computational assumption as is used to prove security of the underlying signature scheme. We show that the following signature schemes have the required properties and thus give rise to secure homomorphic signatures in the standard model:
The scheme of Waters (Eurocrypt 2005), secure under the computational Diffie-Hellman asumption in bilinear groups.
The scheme of Boneh and Boyen (Eurocrypt 2004, J. Cryptology 2008), secure under the q-strong Diffie-Hellman assumption in bilinear groups.
The scheme of Gennaro, Halevi, and Rabin (Eurocrypt 1999), secure under the strong RSA assumption.
The scheme of Hohenberger and Waters (Crypto 2009), secure under the RSA assumption.
Our systems not only allow weaker security assumptions than were previously available for homomorphic signatures in the standard model, but also are secure in a model that allows a stronger adversary than in other proposed schemes.
Our framework also leads to efficient linearly homomorphic signatures that are secure against our stronger adversary under weak assumptions (CDH or RSA) in the random oracle model; all previous proofs of security in the random oracle model break down completely when faced with our stronger adversary.

Privacy homomorphisms, encryption schemes that are also homomorphisms relative to some binary operation, have been studied
for some time, but one may also consider the analogous problem of homomorphic signature schemes. In this paper we introduce
basic definitions of security for homomorphic signature systems, motivate the inquiry with example applications, and describe
several schemes that are homomorphic with respect to useful binary operations. In particular, we describe a scheme that allows
a signature holder to construct the signature on an arbitrarily redacted submessage of the originally signed message. We present
another scheme for signing sets that is homomorphic with respect to both union and taking subsets. Finally, we show that any
signature scheme that is homomorphic with respect to integer addition must be insecure.

Network coding has been shown to improve the capacity and robustness in networks. However, since intermediate nodes modify
packets en-route, integrity of data cannot be checked using traditional MACs and checksums. In addition, network coded systems
are vulnerable to pollution attacks where a single malicious node can flood the network with bad packets and prevent the receiver
from decoding the packets correctly. Signature schemes have been proposed to thwart such attacks, but they tend to be too
slow for online per-packet integrity.
Here we propose a homomorphic MAC which allows checking the integrity of network coded data. Our homomorphic MAC is designed as a drop-in replacement for traditional
MACs (such as HMAC) in systems using network coding.

We construct the first homomorphic signature scheme that is capable of evaluating multivariate polynomials on signed data.
Given the public key and a signed data set, there is an efficient algorithm to produce a signature on the mean, standard deviation,
and other statistics of the signed data. Previous systems for computing on signed data could only handle linear operations.
For polynomials of constant degree, the length of a derived signature only depends logarithmically on the size of the data
set.
Our system uses ideal lattices in a way that is a “signature analogue” of Gentry’s fully homomorphic encryption. Security
is based on hard problems on ideal lattices similar to those in Gentry’s system.

Network coding offers the potential to increase throughput and improve robustness without any centralized control. Unfortunately, network coding is highly susceptible to “pollution attacks” in which malicious nodes modify packets improperly so as to prevent message recovery at the recipient(s); such attacks cannot be prevented using standard end-to-end cryptographic authentication because network coding mandates that intermediate nodes modify data packets in transit.
Specialized “network coding signatures” addressing this problem have been developed in recent years using homomorphic hashing and homomorphic signatures. We contribute to this area in several ways:
We show the first homomorphic signature scheme based on the RSA assumption (in the random oracle model).
We give a homomorphic hashing scheme that is more efficient than existing schemes, and which leads to network coding signatures based on the hardness of factoring (in the standard model).
We describe variants of existing schemes that reduce the communication overhead for moderate-size networks, and improve computational efficiency (in some cases quite dramatically – e.g., we achieve a 20-fold speedup in signature generation at intermediate nodes).
Underlying our techniques is a modified approach to random linear network coding where instead of working in a vector space over a field, we work in a module over the integers (with small coefficients).

Network coding is known to provide improved resilience to packet loss and increased throughput. Unlike traditional routing
techniques, it allows network nodes to perform transformations on packets they receive before transmitting them. For this
reason, packets cannot be authenticated using ordinary digital signatures, which makes it difficult to hedge against pollution
attacks, where malicious nodes inject bogus packets in the network. To address this problem, recent works introduced signature
schemes allowing to sign linear subspaces (namely, verification can be made w.r.t. any vector of that subspace) and which
are well-suited to the network coding scenario. Currently known network coding signatures in the standard model are not homomorphic
in that the signer is forced to sign all vectors of a given subspace at once. This paper describes the first homomorphic network
coding signatures in the standard model: the security proof does not use random oracles and, at the same time, the scheme
allows signing individual vectors on-the-fly and has constant per-packet overhead in terms of signature size. The construction is based on the dual encryption technique
introduced by Waters (Crypto’09) to prove the security of hierarchical identity-based encryption schemes.

We propose a linearly homomorphic signature scheme that authenticates vector subspaces of a given ambient space. Our system has several novel properties not found in previous proposals:
It is the first such scheme that authenticates vectors defined over binary fields; previous proposals could only authenticate vectors with large or growing coefficients.
It is the first such scheme based on the problem of finding short vectors in integer lattices, and thus enjoys the worst-case security guarantees common to lattice-based cryptosystems.
Our scheme can be used to authenticate linear transformations of signed data, such as those arising when computing mean and Fourier transform or in networks that use network coding. Our construction gives an example of a cryptographic primitive — homomorphic signatures over \(\mathbb{F}_2\) — that can be built using lattice methods, but cannot currently be built using bilinear maps or other traditional algebraic methods based on factoring or discrete log type problems.
Security of our scheme (in the random oracle model) is based on a new hard problem on lattices, called k −SIS, that reduces to standard average-case and worst-case lattice problems. Our formulation of the k −SIS problem adds to the “toolbox” of lattice-based cryptography and may be useful in constructing other lattice-based cryptosystems.
As a second application of the new k −SIS tool, we construct an ordinary signature scheme and prove it k-time unforgeable in the standard model assuming the hardness of the k −SIS problem. Our construction can be viewed as “removing the random oracle” from the signatures of Gentry, Peikert, and Vaikuntanathan at the expense of only allowing a small number of signatures.

We present a digital signature scheme based on the computational diculty of integer factorization. The scheme possesses the novel property of being robust against an adaptive chosen-message attack: an adversary who receives signatures for messages of his choice (where each message may be chosen in a way that depends on the signatures of previously chosen messages) can not later forge the signature of even a single additional message. This may be somewhat surprising, since the properties of having forgery being equivalent to factoring and being invulnerable to an adaptive chosen-message attack were considered in the folklore to be contradictory. More generally, we show how to construct a signature scheme with such properties based on the existence of a "claw-free" pair of permutations - a potentially weaker assumption than the intractibility of integer factorization. The new scheme is potentially practical: signing and verifying signatures are reasonably fast, and signatures are compact.

Network Coding is a routing technique where each node may actively modify the received packets before transmitting them.While this departure from passive networks improves throughput and resilience to packet loss it renders transmission susceptible to pollution attacks where nodes can misbehave and change in a malicious way the messages transmitted. Nodes cannot use standard signature schemes to authenticate the modified packets: this would require knowledge of the original sender's signing key. Network coding signature schemes offer a cryptographic solution to this problem. Very roughly, such signatures allow signing vector spaces (or rather bases of such spaces), and these signatures are homomorphic: given signatures on a set of vectors it is possible to create signatures for any linear combination of these vectors. Designing such schemes is a difficult task, and the few existent constructions either rely on random oracles or are rather inefficient. In this paper we introduce two new network coding signature schemes. Both of our schemes are provably secure in the standard model, rely on standard assumptions, and are in the same efficiency class as previous solutions based on random oracles.

Network coding offers increased throughput and improved robustness to random faults in completely decentralized networks. In contrast to traditional routing schemes, however, network coding requires intermediate nodes to modify data packets en route; for this reason, standard signature schemes are inapplicable and it is a challenge to provide resilience to tampering by malicious nodes.
We propose two signature schemes that can be used in conjunction with network coding to prevent malicious modification of data. Our schemes can be viewed as signing linear subspaces in the sense that a signature σ on a subspace V authenticates exactly those vectors in V. Our first scheme is (suitably) homomorphic and has constant public-key size and per-packet overhead. Our second scheme does not rely on random oracles and is based on weaker assumptions.
We also prove a lower bound on the length of signatures for linear subspaces showing that our schemes are essentially optimal in this regard.

On the security notions for homomorphic signatures. Full Version: Cryptology ePrint Archive

- D Catalano
- D Fiore
- L Nizzardo

Online-offline homomorphic signatures for polynomial functions

- K Elkhiyaoui
- M Önen
- R Molva