Generic-Group Delay Functions Require Hidden-Order Groups

Despite the fundamental importance of delay functions, underlying both the classic notion of a time-lock puzzle and the more recent notion of a verifiable delay function, the only known delay function that offers both sufficient structure for realizing these two notions and a realistic level of practicality is the ``iterated squaring\u27\u27 construction of Rivest, Shamir and Wagner. This construction, however, is based on rather strong assumptions in groups of hidden orders, such as the RSA group (which requires a trusted setup) or the class group of an imaginary quadratic number field (which is still somewhat insufficiently explored from the cryptographic perspective). For more than two decades, the challenge of constructing delay functions in groups of known orders, admitting a variety of well-studied instantiations, has eluded the cryptography community. In this work we prove that there are no constructions of generic-group delay functions in cyclic groups of known orders: We show that for any delay function that does not exploit any particular property of the representation of the underlying group, there exists an attacker that completely breaks the function\u27s sequentiality when given the group\u27s order. As any time-lock puzzle and verifiable delay function give rise to a delay function, our result holds for these two notions we well, and explains the lack of success in resolving the above-mentioned long-standing challenge. Moreover, our result holds even if the underlying group is equipped with a $d$-linear map, for any constant $d \geq 2$ (and even for super-constant values of $d$ under certain conditions)

Adaptive Proofs Have Straightline Extractors (in the Random Oracle Model)

Abstract. The concept of adaptive security for proofs of knowledge was recently studied by Bernhard et al. They formalised adaptive security in the ROM and showed that the non-interactive version of the Schnorr protocol obtained using the Fiat-Shamir transformation is not adaptively secure unless the one-more discrete logarithm problem is easy. Their only construction for adaptively secure protocols used the Fischlin transformation [3] which yields protocols with straight-line extractors. In this paper we provide two further key insights. Our main result shows that any adaptively secure protocol must have a straight-line extractor: even the most clever rewinding strategies cannot offer any benefits against adaptive provers. Then, we show that any Fiat-Shamir transformed SIGMA-protocol is not adaptively secure unless a related problem which we call the SIGMA-one-wayness problem is easy. This assumption concerns not just Schnorr but applies to a whole class of SIGMA-protocols including e.g. Chaum-Pedersen and representation proofs. We also prove that SIGMA-one-wayness is hard in the generic group model. Taken together, these results suggest that Fiat-Shamir transformed SIGMA-protocols should not be used in settings where adaptive security is important

Snarky Signatures: Minimal Signatures of Knowledge from Simulation-Extractable SNARKs

We construct a pairing based simulation-extractable SNARK (SE-SNARK) that consists of only 3 group elements and has highly efficient verification. By formally linking SE-SNARKs to signatures of knowledge, we then obtain a succinct signature of knowledge consisting of only 3 group elements. SE-SNARKs enable a prover to give a proof that they know a witness to an instance in a manner which is: (1) succinct - proofs are short and verifier computation is small; (2) zero-knowledge - proofs do not reveal the witness; (3) simulation-extractable - it is only possible to prove instances to which you know a witness, even when you have already seen a number of simulated proofs. We also prove that any pairing based signature of knowledge or SE-NIZK argument must have at least 3 group elements and 2 verification equations. Since our constructions match these lower bounds, we have the smallest size signature of knowledge and the smallest size SE-SNARK possible

Une classification des hypothèses calculatoire dans le modèle du groupe algébrique

International audiencea We give a taxonomy of computational assumptions in the algebraic group model (AGM). We first analyze Boyen's Uber assumption family for bilinear groups and then extend it in several ways to cover assumptions as diverse as Gap Diffie-Hellman and LRSW. We show that in the AGM every member of these families is implied by the q-discrete logarithm (DL) assumption, for some q that depends on the degrees of the polynomials defining the Uber assumption. Using the meta-reduction technique, we then separate (q + 1)-DL from q-DL, which yields a classification of all members of the extended Uber-assumption families. We finally show that there are strong assumptions, such as one-more DL, that provably fall outside our classification, by proving that they cannot be reduced from q-DL even in the AGM

Generically Speeding-Up Repeated Squaring is Equivalent to Factoring: Sharp Thresholds for All Generic-Ring Delay Functions

Despite the fundamental importance of delay functions, repeated squaring in RSA groups (Rivest, Shamir and Wagner \u2796) is the main candidate offering both a useful structure and a realistic level of practicality. Somewhat unsatisfyingly, its sequentiality is provided directly by assumption (i.e., the function is assumed to be a delay function). We prove sharp thresholds on the sequentiality of all generic-ring delay functions relative to an RSA modulus based on the hardness of factoring in the standard model. In particular, we show that generically speeding-up repeated squaring (even with a preprocessing stage and any polynomial number parallel processors) is equivalent to factoring. More generally, based on the (essential) hardness of factoring, we prove that any generic-ring function is in fact a delay function, admitting a sharp sequentiality threshold that is determined by our notion of sequentiality depth. Moreover, we show that generic-ring functions admit not only sharp sequentiality thresholds, but also sharp pseudorandomness thresholds

A Subversion-Resistant SNARK

While succinct non-interactive zero-knowledge arguments of knowledge (zk-SNARKs) are widely studied, the question of what happens when the CRS has been subverted has received little attention. In ASIACRYPT 2016, Bellare, Fuchsbauer and Scafuro showed the first negative and positive results in this direction, proving also that it is impossible to achieve subversion soundness and (even non-subversion) zero knowledge at the same time. On the positive side, they constructed an involved sound and subversion zero-knowledge argument system for NP. We show that Groth\u27s zk-SNARK for \textsc{Circuit-SAT} from EUROCRYPT 2016 can be made computationally knowledge-sound and perfectly composable Sub-ZK with minimal changes. We just require the CRS trapdoor to be extractable and the CRS to be publicly verifiable. To achieve the latter, we add some new elements to the CRS and construct an efficient CRS verification algorithm. We also provide a definitional framework for sound and Sub-ZK SNARKs and describe implementation results of the new Sub-ZK SNARK

Tight Reductions for Diffie-Hellman Variants in the Algebraic Group Model

Fuchsbauer, Kiltz, and Loss~(Crypto\u2718) gave a simple and clean definition of an ¥emph{algebraic group model~(AGM)} that lies in between the standard model and the generic group model~(GGM). Specifically, an algebraic adversary is able to exploit group-specific structures as the standard model while the AGM successfully provides meaningful hardness results as the GGM. As an application of the AGM, they show a tight computational equivalence between the computing Diffie-Hellman~(CDH) assumption and the discrete logarithm~(DL) assumption. For the purpose, they used the square Diffie-Hellman assumption as a bridge, i.e., they first proved the equivalence between the DL assumption and the square Diffie-Hellman assumption, then used the known equivalence between the square Diffie-Hellman assumption and the CDH assumption. In this paper, we provide an alternative proof that directly shows the tight equivalence between the DL assumption and the CDH assumption. The crucial benefit of the direct reduction is that we can easily extend the approach to variants of the CDH assumption, e.g., the bilinear Diffie-Hellman assumption. Indeed, we show several tight computational equivalences and discuss applicabilities of our techniques

The Distinction Between Fixed and Random Generators in Group-Based Assumptions

There is surprisingly little consensus on the precise role of the generator g in group-based assumptions such as DDH. Some works consider g to be a fixed part of the group description, while others take it to be random. We study this subtle distinction from a number of angles. - In the generic group model, we demonstrate the plausibility of groups in which random-generator DDH (resp. CDH) is hard but fixed-generator DDH (resp. CDH) is easy. We observe that such groups have interesting cryptographic applications. - We find that seemingly tight generic lower bounds for the Discrete-Log and CDH problems with preprocessing (Corrigan-Gibbs and Kogan, Eurocrypt 2018) are not tight in the sub-constant success probability regime if the generator is random. We resolve this by proving tight lower bounds for the random generator variants; our results formalize the intuition that using a random generator will reduce the effectiveness of preprocessing attacks. - We observe that DDH-like assumptions in which exponents are drawn from low-entropy distributions are particularly sensitive to the fixed- vs. random-generator distinction. Most notably, we discover that the Strong Power DDH assumption of Komargodski and Yogev (Komargodski and Yogev, Eurocrypt 2018) used for non-malleable point obfuscation is in fact false precisely because it requires a fixed generator. In response, we formulate an alternative fixed-generator assumption that suffices for a new construction of non-malleable point obfuscation, and we prove the assumption holds in the generic group model. We also give a generic group proof for the security of fixed-generator, low-entropy DDH (Canetti, Crypto 1997)

Towards a Classification of Non-interactive Computational Assumptions in Cyclic Groups

We study non-interactive computational intractability assumptions in prime-order cyclic groups. We focus on the broad class of computational assumptions which we call target assumptions where the adversary’s goal is to compute concrete group elements. Our analysis identifies two families of intractability assumptions, the q-Generalized Diffie-Hellman Exponent (q-GDHE) assumptions and the q-Simple Fractional (q-SFrac) assumptions (a natural generalization of the q-SDH assumption), that imply all other target assumptions. These two assumptions therefore serve as Uber assumptions that can underpin all the target assumptions where the adversary has to compute specific group elements. We also study the internal hierarchy among members of these two assumption families. We provide heuristic evidence that both families are necessary to cover the full class of target assumptions. We also prove that having (polynomially many times) access to an adversarial 1-GDHE oracle, which returns correct solutions with non-negligible probability, entails one to solve any instance of the Computational Diffie-Hellman (CDH) assumption. This proves equivalence between the CDH and 1-GDHE assumptions. The latter result is of independent interest. We generalize our results to the bilinear group setting. For the base groups, our results translate nicely and a similar structure of non-interactive computational assumptions emerges. We also identify Uber assumptions in the target group but this requires replacing the q-GDHE assumption with a more complicated assumption, which we call the bilinar gap assumption. Our analysis can assist both cryptanalysts and cryptographers. For cryptanalysts, we propose the q-GDHE and the q-SDH assumptions are the most natural and important targets for cryptanalysis in prime-order groups. For cryptographers, we believe our classification can aid the choice of assumptions underpinning cryptographic schemes and be used as a guide to minimize the overall attack surface that different assumptions expose

Accumulators in (and Beyond) Generic Groups: Non-Trivial Batch Verification Requires Interaction

We prove a tight lower bound on the number of group operations required for batch verification by any generic-group accumulator that stores a less-than-trivial amount of information. Specifically, we show that $\Omega(t \cdot (\lambda / \log \lambda))$ group operations are required for the batch verification of any subset of $t \geq 1$ elements, where $\lambda \in \mathbb{N}$ is the security parameter, thus ruling out non-trivial batch verification in the standard non-interactive manner. Our lower bound applies already to the most basic form of accumulators (i.e., static accumulators that support membership proofs), and holds both for known-order (and even multilinear) groups and for unknown-order groups, where it matches the asymptotic performance of the known bilinear and RSA accumulators, respectively. In addition, it complements the techniques underlying the generic-group accumulators of Boneh, B{ü}nz and Fisch (CRYPTO \u2719) and Thakur (ePrint \u2719) by justifying their application of the Fiat-Shamir heuristic for transforming their interactive batch-verification protocols into non-interactive procedures. Moreover, motivated by a fundamental challenge introduced by Aggarwal and Maurer (EUROCRYPT \u2709), we propose an extension of the generic-group model that enables us to capture a bounded amount of arbitrary non-generic information (e.g., least-significant bits or Jacobi symbols that are hard to compute generically but are easy to compute non-generically). We prove our lower bound within this extended model, which may be of independent interest for strengthening the implications of impossibility results in idealized models