## CryptoDB

### Ronald Cramer

#### Publications

Year
Venue
Title
2021
CRYPTO
In a proof of partial knowledge, introduced by Cramer, Damg{\aa}rd and Schoenmakers (CRYPTO 1994), a prover knowing witnesses for some $k$-subset of $n$ given public statements can convince the verifier of this claim without revealing which $k$-subset. Their solution combines $\Sigma$-protocol theory and linear secret sharing, and achieves linear communication complexity for general $k,n$. Especially the one-out-of-$n$'' case $k=1$ has seen myriad applications during the last decades, e.g., in electronic voting, ring signatures, and confidential transaction systems. In this paper we focus on the discrete logarithm (DL) setting, where the prover claims knowledge of DLs of $k$-out-of-$n$ given elements. Groth and Kohlweiss (EUROCRYPT 2015) have shown how to solve the special case $k=1$ %, yet arbitrary~$n$, with {\em logarithmic} (in $n$) communication, instead of linear as prior work. However, their method takes explicit advantage of $k=1$ and does not generalize to $k>1$. Alternatively, an {\em indirect} approach for solving the considered problem is by translating the $k$-out-of-$n$ relation into a circuit and then applying communication-efficient circuit ZK. Indeed, for the $k=1$ case this approach has been highly optimized, e.g., in ZCash. Our main contribution is a new, simple honest-verifier zero-knowledge proof protocol for proving knowledge of $k$ out of $n$ DLs with {\em logarithmic} communication and {\em for general $k$ and $n$}, without requiring any generic circuit ZK machinery. Our solution puts forward a novel extension of the {\em compressed} $\Sigma$-protocol theory (CRYPTO 2020), which we then utilize to compress a new $\Sigma$-protocol for proving knowledge of $k$-out-of-$n$ DL's down to logarithmic size. The latter $\Sigma$-protocol is inspired by the CRYPTO 1994 approach, but a careful re-design of the original protocol is necessary for the compression technique to apply. Interestingly, {\em even for $k=1$ and general $n$} our approach improves prior {\em direct} approaches as it reduces prover complexity without increasing the communication complexity. Besides the conceptual simplicity, we also identify regimes of practical relevance where our approach achieves asymptotic and concrete improvements, e.g., in proof size and prover complexity, over the generic approach based on circuit-ZK. Finally, we show various extensions and generalizations of our core result. For instance, we extend our protocol to proofs of partial knowledge of Pedersen (vector) commitment openings, and/or to include a proof that the witness satisfies some additional constraint, and we show how to extend our results to non-threshold access structures.
2021
CRYPTO
We show a \emph{lattice-based} solution for commit-and-prove transparent circuit zero-knowledge (ZK) with \emph{polylog-communication}, the \emph{first} not depending on PCPs. We start from \emph{compressed $\Sigma$-protocol theory} (CRYPTO 2020), which is built around basic $\Sigma$-protocols for opening an arbitrary linear form on a long secret vector that is compactly committed to. These protocols are first compressed using a recursive folding-technique'' adapted from Bulletproofs, at the expense of logarithmic rounds. Proving in ZK that the secret vector satisfies a given constraint -- captured by a circuit -- is then by (blackbox) reduction to the linear case, via arithmetic secret-sharing techniques adapted from MPC. Commit-and-prove is also facilitated, i.e., when commitment(s) to the secret vector are created ahead of any circuit-ZK proof. On several platforms (incl.\ DL) this leads to logarithmic communication. Non-interactive versions follow from Fiat-Shamir. This abstract modular theory strongly suggests that it should somehow be supported by a lattice-platform \emph{as well}. However, when going through the motions and trying to establish low communication (on a SIS-platform), a certain significant lack in current understanding of multi-round protocols is exposed. Namely, as opposed to the DL-case, the basic $\Sigma$-protocol in question typically has \emph{poly-small challenge} space. Taking into account the compression-step -- which yields \emph{non-constant} rounds -- and the necessity for parallelization to reduce error, there is no known tight result that the compound protocol admits an efficient knowledge extractor. We resolve the state of affairs here by a combination of two novel results which are fully general and of independent interest. The first gives a tight analysis of efficient knowledge extraction in case of non-constant rounds combined with poly-small challenge space, whereas the second shows that parallel repetition indeed forces rapid decrease of knowledge error. Moreover, in our present context, arithmetic secret sharing is not defined over a large finite field but over a quotient of a number ring and this forces our careful adaptation of how the linearization techniques are deployed. We develop our protocols in an abstract framework that is conceptually simple and can be flexibly instantiated. In particular, the framework applies to arbitrary rings and norms.
2021
CRYPTO
The current paper studies information-theoretically secure multiparty computation (MPC) over rings $\Z/p^{\ell}\Z$. This is a follow-up research of recent work on MPC over rings $\Z/p^{\ell}\Z$. In the work of \cite[TCC2019]{tcc}, a protocol based on the Shamir secret sharing over $\Z/p^{\ell}\Z$ was presented. As in the field case, its limitation is that the share size has to grow as the number of players increases. Then several MPC protocols were developed in \cite[Asiacrypt 2020]{asiacrypt} to overcome this limitation. However, the MPC protocols in \cite[Asiacrypt 2020]{asiacrypt} suffer from several drawbacks: (i) the offline multiplication gate has super-linear communication complexity; (ii) the share size is doubled for the most important case, namely over $\Z/2^{\ell}\Z$ due to infeasible lifting of self-orthogonal codes from fields to rings; (iii) most importantly, the BGW model could not be applied via the secret sharing given in \cite[Asiacrypt 2020]{asiacrypt} due to lack of strong multiplication. Our contribution in this paper is three fold. Firstly, we overcome all the drawbacks in \cite{tcc,asiacrypt} mentioned above. Secondly, we establish an arithmetic secret sharing with strong multiplication, which is the most important primitive in the BGW model. Thirdly, we lift Reverse Multiplication Friendly Embeddings (RMFE) from fields to rings, with same (linear) complexity. Note that RMFE has become a standard technique for amortized communication complexity in MPC, as in \cite[CRYPTO'18]{crypto2018} and \cite[CRYPTO'19]{dn19}. To obtain our theoretical results, we use the existence of lifts of curves over rings, then use the known results stating that Riemann-Roch spaces are free modules. To make our scheme practical, we start from good algebraic geometry codes over finite fields obtained from existing computational techniques. Then we present, and implement, an efficient algorithm to Hensel-lift the generating matrix of the code, such that the multiplicative conditions are preserved over rings. Existence of this specific lift is guaranteed by the previous theory. On the other hand, a random lifting of codes over from fields to Galois rings does not preserve multiplicativity in general. (Notice that our indirect method is motivated by the fact that, following the theory instead, would require to preprocess'' the curve under a form with smooth" equations, in particular with many variables, before lifting it. But computing on these objects over rings is out of the scope of existing research). Finally we provide efficient elementary methods for sharing and (robust) reconstruction of secrets over rings. As a result, arithmetic secret sharing over $\Z/p^{\ell}\Z$ with strong multiplication can be efficiently constructed and practically applied.
2021
ASIACRYPT
Lai et al. (CCS 2019) have shown how Bulletproof’s arithmetic circuit zero-knowledge protocol (Bootle et al., EUROCRYPT 2016 and B{\"u}nz et al., S\&P 2018) can be generalized to work for bilinear group arithmetic circuits directly, i.e., without requiring these circuits to be translated into arithmetic circuits. In a nutshell, a bilinear group arithmetic circuit is a standard arithmetic circuit augmented with special gates capturing group exponentiations or pairings. Such circuits are highly relevant, e.g., in the context of zero-knowledge statements over pairing-based languages. As expressing these special gates in terms of a standard arithmetic circuit results in a significant overhead in circuit size, an approach to zero-knowledge via standard arithmetic circuits may incur substantial additional costs. The approach due to Lai et al. shows how to avoid this by integrating additional zero-knowledge techniques into the Bulletproof framework so as to handle the special gates very efficiently. We take a different approach by generalizing {\em Compressed $\Sigma$-Protocol Theory} (CRYPTO 2020) from arithmetic circuit relations to bilinear group arithmetic circuit relations. Besides its conceptual simplicity, our approach has the practical advantage of reducing the communication costs of Lai et al.'s protocol by roughly a multiplicative factor $3$. Finally, we show an application of our results which may be of independent interest. We construct the first $k$-out-of-$n$ threshold signature scheme (TSS) that allows for transparent setup {\em and} that yields threshold signatures of size logarithmic in $n$. The threshold signature hides the identities of the $k$ signers and the threshold $k$ can be dynamically chosen at aggregation time.
2021
ASIACRYPT
In 2016, Guruswami and Wootters showed Shamir's secret-sharing scheme defined over an extension field has a regenerating property. Namely, we can compress each share to an element of the base field by applying a linear form, such that the secret is determined by a linear combination of the compressed shares. Immediately it seemed like an application to improve the complexity of unconditionally secure multiparty computation must be imminent; however, thus far, no result has been published. We present the first application of regenerating codes to MPC, and show that its utility lies in reducing the number of rounds. Concretely, we present a protocol that obliviously evaluates a depth-$d$ arithmetic circuit in $d + O(1)$ rounds, in the amortized setting of parallel evaluations, with $o(n^2)$ ring elements communicated per multiplication. Our protocol makes use of function-dependent preprocessing, and is secure against the maximal adversary corrupting $t < n/2$ parties. All existing approaches in this setting have complexity $\Omega(n^2)$. Moreover, we extend some of the theory on regenerating codes to Galois rings. It was already known that the repair property of MDS codes over fields can be fully characterized in terms of its dual code. We show this characterization extends to linear codes over Galois rings, and use it to show the result of Guruswami and Wootters also holds true for Shamir's scheme over Galois rings.
2020
EUROCRYPT
A {\em blackbox} secret sharing (BBSS) scheme works in exactly the same way for all finite Abelian groups $G$; it can be instantiated for any such group $G$ and {\em only} black-box access to its group operations and to random group elements is required. A secret is a single group element and each of the $n$ players' shares is a vector of such elements. Share-computation and secret-reconstruction is by integer linear combinations. These do not depend on $G$, and neither do the privacy and reconstruction parameters $t,r$. This classical, fundamental primitive was introduced by Desmedt and Frankel (CRYPTO 1989) in their context of threshold cryptography.'' The expansion factor is the total number of group elements in a full sharing divided by $n$. For threshold BBSS with $t$-privacy ($1\leq t \leq n-1$), $t+1$-reconstruction and arbitrary $n$, constructions with minimal expansion $O(\log n)$ exist (CRYPTO 2002, 2005). These results are firmly rooted in number theory; each makes (different) judicious choices of orders in number fields admitting a vector of elements of very large length (in the number field degree) whose corresponding Vandermonde-determinant is sufficiently controlled so as to enable BBSS by a suitable adaptation of Shamir's scheme. Alternative approaches generally lead to very large expansion. The state of the art of BBSS has not changed for the last 15 years. Our contributions are two-fold. (1) We introduce a novel, nontrivial, effective construction of BBSS based on {\em coding theory} instead of number theory. For threshold-BBSS we also achieve minimal expansion factor $O(\log n)$. (2) Our method is more versatile. Namely, we show, for the first time, BBSS that is {\em near-threshold}, i.e., $r-t$ is an arbitrarily small constant fraction of $n$, {\em and} that has expansion factor~$O(1)$, i.e., individual share-vectors of {\em constant} length (asymptotically expansionless''). Threshold can be concentrated essentially freely across full range. We also show expansion is minimal for near-threshold and that such BBSS cannot be attained by previous methods. Our general construction is based on a well-known mathematical principle, the local-global principle. More precisely, we first construct BBSS over local rings through either Reed-Solomon or algebraic geometry codes. We then glue'' these schemes together in a dedicated manner to obtain a global secret sharing scheme, i.e., defined over the integers, which, as we finally prove using novel insights, has the desired BBSS properties. Though our main purpose here is advancing BBSS for its own sake, we also briefly address possible protocol applications.
2020
CRYPTO
Sigma-Protocols provide a well-understood basis for secure algorithmics. Recently, Bulletproofs (Bootle et al., EUROCRYPT 2016, and Bünz et al., S&P 2018) have been proposed as a drop-in replacement in case of zero-knowledge (ZK) for arithmetic circuits, achieving logarithmic communication instead of linear. Its pivot is an ingenious, logarithmic-size proof of knowledge BP for certain quadratic relations. However, reducing ZK for general relations to it forces a somewhat cumbersome reinvention'' of cryptographic protocol theory. We take a rather different viewpoint and reconcile Bulletproofs with Sigma-Protocol Theory such that (a) simpler circuit ZK is developed within established theory, while (b) achieving exactly the same logarithmic communication. The natural key here is linearization. First, we repurpose BPs as a blackbox compression mechanism for standard Sigma-Protocols handling ZK proofs of general linear relations (on compactly committed secret vectors); our pivot. Second, we reduce the case of general nonlinear relations to blackbox applications of our pivot via a novel variation on arithmetic secret sharing based techniques for Sigma-Protocols (Cramer et al., ICITS 2012). Orthogonally, we enhance versatility by enabling scenarios not previously addressed, e.g., when a secret input is dispersed across several commitments. Standard implementation platforms leading to logarithmic communication follow from a Discrete-Log assumption or a generalized Strong-RSA assumption. Also, under a Knowledge-of-Exponent Assumption (KEA) communication drops to constant, as in ZK-SNARKS. All in all, our theory should more generally be useful for modular (plug & play'') design of practical cryptographic protocols; this is further evidenced by our separate work (2020) on proofs of partial knowledge.
2020
TCC
Since the mid 2000s, asymptotically-good strongly-multiplicative linear (ramp) secret sharing schemes over a fixed finite field have turned out as a central theoretical primitive in numerous constant-communication-rate results in multi-party cryptographic scenarios, and, surprisingly, in two-party cryptography as well. Known constructions of this most powerful class of arithmetic secret sharing schemes all rely heavily on algebraic geometry (AG), i.e., on dedicated AG codes based on asymptotically good towers of algebraic function fields defined over finite fields. It is a well-known open question since the first (explicit) constructions of such schemes appeared in CRYPTO 2006 whether the use of heavy machinery'' can be avoided here. i.e., the question is whether the mere existence of such schemes can also be proved by elementary'' techniques only (say, from classical algebraic coding theory), even disregarding effective construction. So far, there is no progress. In this paper we show the theoretical result that, (1) {\em no matter whether this open question has an affirmative answer or not}, these schemes {\em can} be constructed explicitly by {\em elementary algorithms} defined in terms of basic algebraic coding theory. This pertains to all relevant operations associated to such schemes, including, notably, the generation of an instance for a given number of players $n$, as well as error correction in the presence of corrupt shares. We further show that (2) the algorithms are {\em quasi-linear time} (in $n$); this is (asymptotically) significantly more efficient than the known constructions. That said, the {\em analysis} of the mere termination of these algorithms {\em does} still rely on algebraic geometry, in the sense that it requires blackbox application'' of suitable {\em existence} results for these schemes. Our method employs a nontrivial, novel adaptation of a classical (and ubiquitous) paradigm from coding theory that enables transformation of {\em existence} results on asymptotically good codes into {\em explicit construction} of such codes via {\em concatenation}, at some constant loss in parameters achieved. In a nutshell, our generating idea is to combine a cascade of explicit but asymptotically-bad-yet-good-enough schemes'' with an asymptotically good one in such a judicious way that the latter can be selected with exponentially small number of players in that of the compound scheme. This opens the door to efficient, elementary exhaustive search. In order to make this work, we overcome a number of nontrivial technical hurdles. Our main handles include a novel application of the recently introduced notion of Reverse Multiplication-Friendly Embeddings (RMFE) from CRYPTO 2018, as well as a novel application of a natural variant in arithmetic secret sharing from EUROCRYPT 2008.
2020
ASIACRYPT
We study information-theoretic multiparty computation (MPC) protocols over rings Z/p^k Z that have good asymptotic communication complexity for a large number of players. An important ingredient for such protocols is arithmetic secret sharing, i.e., linear secret-sharing schemes with multiplicative properties. The standard way to obtain these over fields is with a family of linear codes C, such that C, $C^\perp$ and C^2 are asymptotically good (strongly multiplicative). For our purposes here it suffices if the square code C^2 is not the whole space, i.e., has codimension at least 1 (multiplicative). Our approach is to lift such a family of codes defined over a finite field F to a Galois ring, which is a local ring that has F as its residue field and that contains Z/p^k Z as a subring, and thus enables arithmetic that is compatible with both structures. Although arbitrary lifts preserve the distance and dual distance of a code, as we demonstrate with a counterexample, the multiplicative property is not preserved. We work around this issue by showing a dedicated lift that preserves \emph{self-orthogonality} (as well as distance and dual distance), for p > 2. Self-orthogonal codes are multiplicative, therefore we can use existing results of asymptotically good self-dual codes over fields to obtain arithmetic secret sharing over Galois rings. For p = 2 we obtain multiplicativity by using existing techniques of secret-sharing using both C and $C^\perp$, incurring a constant overhead. As a result, we obtain asymptotically good arithmetic secret-sharing schemes over Galois rings. With these schemes in hand, we extend existing field-based MPC protocols to obtain MPC over Z/p^k Z, in the setting of a submaximal adversary corrupting less than a fraction 1/2 - \varepsilon of the players, where \varepsilon > 0 is arbitrarily small. We consider 3 different corruption models, and obtain O(n) bits communicated per multiplication for both passive security and active security with abort. For full security with guaranteed output delivery we use a preprocessing model and get O(n) bits per multiplication in the online phase and O(n log n) bits per multiplication in the offline phase. Thus, we obtain true linear bit complexities, without the common assumption that the ring size depends on the number of players.
2019
TCC
At CRYPTO 2018, Cramer et al. introduced a secret-sharing based protocol called SPD$\mathbb {Z}_{2^k}$ that allows for secure multiparty computation (MPC) in the dishonest majority setting over the ring of integers modulo $2^k$, thus solving a long-standing open question in MPC about secure computation over rings in this setting. In this paper we study this problem in the information-theoretic scenario. More specifically, we ask the following question: Can we obtain information-theoretic MPC protocols that work over rings with comparable efficiency to corresponding protocols over fields? We answer this question in the affirmative by presenting an efficient protocol for robust Secure Multiparty Computation over $\mathbb {Z}/p^{k}\mathbb {Z}$ (for any prime p and positive integer k) that is perfectly secure against active adversaries corrupting a fraction of at most 1/3 players, and a robust protocol that is statistically secure against an active adversary corrupting a fraction of at most 1/2 players.
2018
CRYPTO
A fundamental and widely-applied paradigm due to Franklin and Yung (STOC 1992) on Shamir-secret-sharing based general n-player MPC shows how one may trade the adversary thresholdt against amortized communication complexity, by using a so-called packed version of Shamir’s scheme. For e.g. the BGW-protocol (with active security), this trade-off means that if $t + 2k -2 < n/3$ t+2k-2<n/3, then kparallel evaluations of the same arithmetic circuit on different inputs can be performed at the overall cost corresponding to a single BGW-execution.In this paper we propose a novel paradigm for amortized MPC that offers a different trade-off, namely with the size of the field of the circuit which is securely computed, instead of the adversary threshold. Thus, unlike the Franklin-Yung paradigm, this leaves the adversary threshold unchanged. Therefore, for instance, this paradigm may yield constructions enjoying the maximal adversary threshold $\lfloor (n-1)/3 \rfloor$ ⌊(n-1)/3⌋ in the BGW-model (secure channels, perfect security, active adversary, synchronous communication).Our idea is to compile an MPC for a circuit over an extension field to a parallel MPC of the same circuit but with inputs defined over its base field and with the same adversary threshold. Key technical handles are our notion of reverse multiplication-friendly embeddings (RMFE) and our proof, by algebraic-geometric means, that these are constant-rate, as well as efficient auxiliary protocols for creating “subspace-randomness” with good amortized complexity. In the BGW-model, we show that the latter can be constructed by combining our tensored-up linear secret sharing with protocols based on hyper-invertible matrices á la Beerliova-Hirt (or variations thereof). Along the way, we suggest alternatives for hyper-invertible matrices with the same functionality but which can be defined over a large enough constant size field, which we believe is of independent interest.As a demonstration of the merits of the novel paradigm, we show that, in the BGW-model and with an optimal adversary threshold $\lfloor (n-1)/3 \rfloor$ ⌊(n-1)/3⌋, it is possible to securely compute a binary circuit with amortized complexity O(n) of bits per gate per instance. Known results would give $n \log n$ nlogn bits instead. By combining our result with the Franklin-Yung paradigm, and assuming a sub-optimal adversary (i.e., an arbitrarily small $\epsilon >0$ ϵ>0 fraction below 1/3), this is improved to O(1) bits instead of O(n).
2018
CRYPTO
Most multi-party computation protocols allow secure computation of arithmetic circuits over a finite field, such as the integers modulo a prime. In the more natural setting of integer computations modulo $2^{k}$, which are useful for simplifying implementations and applications, no solutions with active security are known unless the majority of the participants are honest.We present a new scheme for information-theoretic MACs that are homomorphic modulo $2^k$, and are as efficient as the well-known standard solutions that are homomorphic over fields. We apply this to construct an MPC protocol for dishonest majority in the preprocessing model that has efficiency comparable to the well-known SPDZ protocol (Damgård et al., CRYPTO 2012), with operations modulo $2^k$ instead of over a field. We also construct a matching preprocessing protocol based on oblivious transfer, which is in the style of the MASCOT protocol (Keller et al., CCS 2016) and almost as efficient.
2017
EUROCRYPT
2017
EUROCRYPT
2016
EUROCRYPT
2015
EPRINT
2015
EPRINT
2015
TCC
2015
EUROCRYPT
2014
EPRINT
2014
EPRINT
2014
EPRINT
2014
JOFC
2012
PKC
2011
CRYPTO
2011
EUROCRYPT
2010
TCC
2009
CRYPTO
2009
CRYPTO
2008
EUROCRYPT
2008
EUROCRYPT
2008
EPRINT
Consider an abstract storage device $\Sigma(\G)$ that can hold a single element $x$ from a fixed, publicly known finite group $\G$. Storage is private in the sense that an adversary does not have read access to $\Sigma(\G)$ at all. However, $\Sigma(\G)$ is non-robust in the sense that the adversary can modify its contents by adding some offset $\Delta \in \G$. Due to the privacy of the storage device, the value $\Delta$ can only depend on an adversary's {\em a priori} knowledge of $x$. We introduce a new primitive called an {\em algebraic manipulation detection} (AMD) code, which encodes a source $s$ into a value $x$ stored on $\Sigma(\G)$ so that any tampering by an adversary will be detected, except with a small error probability $\delta$. We give a nearly optimal construction of AMD codes, which can flexibly accommodate arbitrary choices for the length of the source $s$ and security level $\delta$. We use this construction in two applications: \begin{itemize} \item We show how to efficiently convert any linear secret sharing scheme into a {\em robust secret sharing scheme}, which ensures that no \emph{unqualified subset} of players can modify their shares and cause the reconstruction of some value $s'\neq s$. \item We show how how to build nearly optimal {\em robust fuzzy extractors} for several natural metrics. Robust fuzzy extractors enable one to reliably extract and later recover random keys from noisy and non-uniform secrets, such as biometrics, by relying only on {\em non-robust public storage}. In the past, such constructions were known only in the random oracle model, or required the entropy rate of the secret to be greater than half. Our construction relies on a randomly chosen common reference string (CRS) available to all parties. \end{itemize}
2007
ASIACRYPT
2007
CRYPTO
2007
EUROCRYPT
2007
EUROCRYPT
2006
CRYPTO
2006
CRYPTO
2006
EPRINT
Designing public key encryption schemes withstanding chosen ciphertext attacks, which is the highest security level for such schemes, is generally perceived as a delicate and intricate task, and for good reason. In the standard model, there are essentially three well-known but quite involved approaches. This state of affairs is to be contrasted with the situation for semantically secure encryption schemes, a much weaker security notion that only guarantees security in the absence of active attack but that appears to be much easier to fulfill, both conceptually and practically. Thus, the boundary between passive attack and active attack seems to make up the dividing line between which security levels are relatively easily achieved and which are not. Our contributions are two-fold. First, we show a simple, efficient black-box construction of a public key encryption scheme withstanding chosen ciphertext attack from any given semantically secure one. Our scheme is $q$-bounded in the sense that security is only guaranteed if the adversary makes at most $q$ adaptive chosen ciphertext queries. Here, $q$ is an arbitrary polynomial that is fixed in advance in the key-generation. Our work thus shows that whether or not the number of active, adversarial queries is known in advance is the dividing line, and not passive versus active attack. In recent work, Gertner, Malkin and Myers show that such black-box reductions are impossible if instead $q$ is a polynomial that only depends on the adversary. Thus, in a sense, our result appears to be the best black-box result one can hope for. Second, we give a non-blackbox reduction from bounded chosen ciphertext security to semantic security where the length of the public/secret keys and ciphertexts drops from quadratic to linear in $q$, compared to our black-box construction. This latter scheme, however, is only of theoretical interest as it uses general NP-reductions, and our blackbox construction is in fact much more practical.
2005
CRYPTO
2005
CRYPTO
2005
TCC
2004
TCC
2004
EPRINT
Error correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, we study the connections between codes, matroids, and a special class of secret sharing schemes: multiplicative linear secret sharing schemes. Such schemes are known to enable multi-party computation protocols secure against general (non-threshold) adversaries. Two open problems related to the complexity of multiplicative LSSSs are considered in this paper. The first one deals with strongly multiplicative LSSSs. As opposed to the case of multiplicative LSSSs, it is not known whether there is an efficient method to transform an LSSS into a strongly multiplicative LSSS for the same access structure with a polynomial increase of the complexity. We prove a property of strongly multiplicative LSSSs that could be useful in solving this problem. Namely, using a suitable generalization of the well-known Berlekamp-Welch decoder, we show that all strongly multiplicative LSSSs enable efficient reconstruction of a shared secret in the presence of malicious faults. The second one is to characterize the access structures of ideal multiplicative LSSSs. Specifically, we wonder whether all self-dual vector space access structures are in this situation. By the aforementioned connection, this in fact constitutes an open problem about matroid theory, since it can be re-stated in terms of representability of identically self-dual matroids by self-dual codes. We introduce a new concept, the flat-partition, that provides a useful classification of identically self-dual matroids. Uniform identically self-dual matroids, which are known to be representable by self-dual codes, form one of the classes. We prove that this property also holds for the family of matroids that, in a natural way, is the next class in the above classification: the identically self-dual bipartite matroids.
2003
EUROCRYPT
2003
EPRINT
Secure multi-party computation (MPC) is an active research area, and a wide range of literature can be found nowadays suggesting improvements and generalizations of existing protocols in various directions. However, all current techniques for secure MPC apply to functions that are represented by (boolean or arithmetic) circuits over finite {\em fields}. We are motivated by two limitations of these techniques: {\sc Generality.} Existing protocols do not apply to computation over more general algebraic structures (except via a brute-force simulation of computation in these structures). {\sc Efficiency.} The best known {\em constant-round} protocols do not efficiently scale even to the case of large finite fields. Our contribution goes in these two directions. First, we propose a basis for unconditionally secure MPC over an arbitrary finite {\em ring}, an algebraic object with a much less nice structure than a field, and obtain efficient MPC protocols requiring only a {\em black-box access} to the ring operations and to random ring elements. Second, we extend these results to the constant-round setting, and suggest efficiency improvements that are relevant also for the important special case of fields. We demonstrate the usefulness of the above results by presenting a novel application of MPC over (non-field) rings to the round-efficient secure computation of the maximum function.
2002
ASIACRYPT
2002
CRYPTO
2002
EUROCRYPT
2002
EPRINT
A {\em black-box} secret sharing scheme for the threshold access structure $T_{t,n}$ is one which works over any finite Abelian group $G$. Briefly, such a scheme differs from an ordinary linear secret sharing scheme (over, say, a given finite field) in that distribution matrix and reconstruction vectors are defined over the integers and are designed {\em independently} of the group $G$ from which the secret and the shares are sampled. This means that perfect completeness and perfect privacy are guaranteed {\em regardless} of which group $G$ is chosen. We define the black-box secret sharing problem as the problem of devising, for an arbitrary given $T_{t,n}$, a scheme with minimal expansion factor, i.e., where the length of the full vector of shares divided by the number of players $n$ is minimal. Such schemes are relevant for instance in the context of distributed cryptosystems based on groups with secret or hard to compute group order. A recent example is secure general multi-party computation over black-box rings. In 1994 Desmedt and Frankel have proposed an elegant approach to the black-box secret sharing problem based in part on polynomial interpolation over cyclotomic number fields. For arbitrary given $T_{t,n}$ with $0<t<n-1$, the expansion factor of their scheme is $O(n)$. This is the best previous general approach to the problem. Using low degree integral extensions of the integers over which there exists a pair of sufficiently large Vandermonde matrices with co-prime determinants, we construct, for arbitrary given $T_{t,n}$ with $0<t<n-1$ , a black-box secret sharing scheme with expansion factor $O(\log n)$, which we show is minimal.
2001
CRYPTO
2001
CRYPTO
2001
EUROCRYPT
2001
EPRINT
A new public key encryption scheme, along with several variants, is proposed and analyzed. The scheme and its variants are quite practical, and are proved secure against adaptive chosen ciphertext attack under standard intractability assumptions. These appear to be the first public-key encryption schemes in the literature that are simultaneously practical and provably secure.
2001
EPRINT
We present several new and fairly practical public-key encryption schemes and prove them secure against adaptive chosen ciphertext attack. One scheme is based on Paillier's Decision Composite Residuosity (DCR) assumption, while another is based in the classical Quadratic Residuosity (QR) assumption. The analysis is in the standard cryptographic model, i.e., the security of our schemes does not rely on the Random Oracle model. We also introduce the notion of a universal hash proof system. Essentially, this is a special kind of non-interactive zero-knowledge proof system for an NP language. We do not show that universal hash proof systems exist for all NP languages, but we do show how to construct very efficient universal hash proof systems for a general class of group-theoretic language membership problems. Given an efficient universal hash proof system for a language with certain natural cryptographic indistinguishability properties, we show how to construct an efficient public-key encryption schemes secure against adaptive chosen ciphertext attack in the standard model. Our construction only uses the universal hash proof systemas a primitive: no other primitives are required, although even more efficient encryption schemes can be obtained by using hash functions with appropriate collision-resistance properties. We show how to construct efficient universal hash proof systems for languages related to the DCR and QR assumptions. From these we get corresponding public-key encryption schemes that are secure under these assumptions. We also show that the Cramer-Shoup encryption scheme (which up until now was the only practical encryption scheme that could be proved secure against adaptive chosen ciphertext attack under a reasonable assumption, namely, the Decision Diffie-Hellman assumption) is also a special case of our general theory.
2000
EUROCRYPT
2000
PKC
2000
EPRINT
We show that verifiable secret sharing (VSS) and secure multi-party computation (MPC) among a set of $n$ players can efficiently be based on {\em any} linear secret sharing scheme (LSSS) for the players, provided that the access structure of the LSSS allows MPC or VSS at all. Because an LSSS neither guarantees reconstructability when some shares are false, nor verifiability of a shared value, nor allows for the multiplication of shared values, an LSSS is an apparently much weaker primitive than VSS or MPC. Our approach to secure MPC is generic and applies to both the in\-for\-ma\-tion-theoretic and the cryptographic setting. The construction is based on 1) a formalization of the special multiplicative property of an LSSS that is needed to perform a multiplication on shared values, 2) an efficient generic construction to obtain from any LSSS a multiplicative LSSS for the same access structure, and 3) an efficient generic construction to build verifiability into every LSSS (always assuming that the adversary structure allows for MPC or VSS at all). The protocols are efficient. In contrast to all previous information-theo\-re\-ti\-cal\-ly secure protocols, the field size is not restricted (e.g, to be greater than $n$). Moreover, we exhibit adversary structures for which our protocols are polynomial in $n$ while all previous approaches to MPC for non-threshold adversaries provably have super-polynomial complexity.
2000
EPRINT
We first study the problem of doing Verifiable Secret Sharing (VSS) information theoretically secure for a general access structure. We do it in the model where private channels between players and a broadcast channel is given, and where an active, adaptive adversary can corrupt any set of players not in the access structure. In particular, we consider the complexity of protocols for this problem, as a function of the access structure and the number of players. For all access structures where VSS is possible at all, we show that, up to a polynomial time black-box reduction, the complexity of adaptively secure VSS is the same as that of ordinary secret sharing (SS), where security is only required against a passive, static adversary. Previously, such a connection was only known for linear secret sharing and VSS schemes. We then show an impossibility result indicating that a similar equivalence does not hold for Multiparty Computation (MPC): we show that even if protocols are given black-box access for free to an idealized secret sharing scheme secure for the access structure in question, it is not possible to handle all relevant access structures efficiently, not even if the adversary is passive and static. In other words, general MPC can only be black-box reduced efficiently to secret sharing if extra properties of the secret sharing scheme used (such as linearity) are assumed.
2000
EPRINT
We initiate the investigation of the class of relations that admit extremely efficient perfect zero knowledge proofs of knowledge: constant number of rounds, communication linear in the length of the statement and the witness, and negligible knowledge error. In its most general incarnation, our result says that for relations that have a particular three-move honest-verifier zero-knowledge (HVZK) proof of knowledge, and which admit a particular three-move HVZK proof of knowledge for an associated commitment relation, perfect zero knowledge (against a general verifier) can be achieved essentially for free, even when proving statements on several instances combined under under monotone function composition. In addition, perfect zero-knowledge is achieved with an optimal 4-moves. Instantiations of our main protocol lead to efficient perfect ZK proofs of knowledge of discrete logarithms and RSA-roots, or more generally, $q$-one-way group homomorphisms. None of our results rely on intractability assumptions.
2000
EPRINT
We introduce a new approach to multiparty computation (MPC) basing it on homomorphic threshold crypto-systems. We show that given keys for any sufficiently efficient system of this type, general MPC protocols for $n$ players can be devised which are secure against an active adversary that corrupts any minority of the players. The total number of bits sent is $O(nk|C|)$, where $k$ is the security parameter and $|C|$ is the size of a (Boolean) circuit computing the function to be securely evaluated. An earlier proposal by Franklin and Haber with the same complexity was only secure for passive adversaries, while all earlier protocols with active security had complexity at least quadratic in $n$. We give two examples of threshold cryptosystems that can support our construction and lead to the claimed complexities.
1999
EUROCRYPT
1999
EPRINT
We describe and analyze a new digital signature scheme. The new scheme is quite efficient, does not require the the signer to maintain any state, and can be proven secure against adaptive chosen message attack under a reasonable intractability assumption, the so-called Strong RSA Assumption. Moreover, a hash function can be incorporated into the scheme in such a way that it is also secure in the random oracle model under the standard RSA Assumption.
1998
CRYPTO
1998
CRYPTO
1998
EPRINT
A new public key cryptosystem is presented that is provably secure against adaptive chosen ciphertext attack. The scheme is quite practical, and the proof of security relies only on standard intractability assumptions.
1997
EUROCRYPT
1997
EUROCRYPT
1996
CRYPTO
1996
EUROCRYPT
1996
EPRINT
Assume we are given a language L with an honest verifier perfect zero-knowledge proof system. Assume also that the proof system is an Arthur-Merlin game with at most 3 moves. The class of such languages includes all random self-reducible language, and also any language with a perfect zero-knowledge non-interactive proof. We show that such a language satisfies a certain closure property, namely that languages constructed from L by applying certain monotone functions to statements on membership in L have perfect zero-knowledge proof systems. The new set of languages we can build includes L itself, but also for example languages consisting of n words of which at least t are in L. A similar closure property is shown to hold for the complement of L and for statistical zero-knowledge. The property we need for the monotone functions used to build the new languages is that there are efficient secret sharing schemes for their associated access structures. This includes (but is not necessarily limited to) all monotone functions with polynomial size monotone formulas.
1996
EPRINT
We present a zero-knowledge proof system for any NP language L, which allows showing that x is in L using communication corresponding to $O(|x| sup c)+k$ bit commitments, with error probability $2 sup -k$, and where c is a constant depending only on L. The proof can be based on any bit commitment scheme with a particular set of properties. We suggest an efficient implementation based on factoring. The protocol allows showing that a Boolean formula of size n is satisfiable, with error probability $2 sup -n$, using O(n) commitments. This is the first protocol for SAT that is linear in this sense.<br> [The rest of the abstract was truncated and appears below -- the library.]
1995
CRYPTO
1994
CRYPTO
1993
EUROCRYPT

#### Program Committees

TCC 2012 (Program chair)
TCC 2011
PKC 2008 (Program chair)
Eurocrypt 2005 (Program chair)
Crypto 2004
TCC 2004
Crypto 2003
Eurocrypt 2003
Asiacrypt 2001
Eurocrypt 2000
Crypto 2000
Crypto 1999