53 research outputs found

    Fiat-Shamir: From Practice to Theory, Part II (NIZK and Correlation Intractability from Circular-Secure FHE)

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    We construct non-interactive zero-knowledge (NIZK) arguments for NP\mathsf{NP} from any circular-secure fully homomorphic encryption (FHE) scheme. In particular, we obtain such NIZKs under a circular-secure variant of the learning with errors (LWE) problem while only assuming a standard (poly/negligible) level of security. Our construction can be modified to obtain NIZKs which are either: (1) statistically zero-knowledge arguments in the common random string model or (2) statistically sound proofs in the common reference string model. We obtain our result by constructing a new correlation-intractable hash family [Canetti, Goldreich, and Halevi, JACM~\u2704] for a large class of relations, which suffices to apply the Fiat-Shamir heuristic to specific 3-message proof systems that we call ``trapdoor ÎŁ\Sigma-protocols.\u27\u27 In particular, assuming circular secure FHE, our hash function hh ensures that for any function ff of some a-priori bounded circuit size, it is hard to find an input xx such that h(x)=f(x)h(x)=f(x). This continues a recent line of works aiming to instantiate the Fiat-Shamir methodology via correlation intractability under progressively weaker and better-understood assumptions. Another consequence of our hash family construction is that, assuming circular-secure FHE, the classic quadratic residuosity protocol of [Goldwasser, Micali, and Rackoff, SICOMP~\u2789] is not zero knowledge when repeated in parallel. We also show that, under the plain LWE assumption (without circularity), our hash family is a universal correlation intractable family for general relations, in the following sense: If there exists any hash family of some description size that is correlation-intractable for general (even inefficient) relations, then our specific construction (with a comparable size) is correlation-intractable for general (efficiently verifiable) relations

    Correlation-Intractable Hash Functions via Shift-Hiding

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    A hash function family H\mathcal{H} is correlation intractable for a tt-input relation R\mathcal{R} if, given a random function hh chosen from H\mathcal{H}, it is hard to find x1,…,xtx_1,\ldots,x_t such that R(x1,…,xt,h(x1),…,h(xt))\mathcal{R}(x_1,\ldots,x_t,h(x_1),\ldots,h(x_t)) is true. Among other applications, such hash functions are a crucial tool for instantiating the Fiat-Shamir heuristic in the plain model, including the only known NIZK for NP based on the learning with errors (LWE) problem (Peikert and Shiehian, CRYPTO 2019). We give a conceptually simple and generic construction of single-input CI hash functions from shift-hiding shiftable functions (Peikert and Shiehian, PKC 2018) satisfying an additional one-wayness property. This results in a clean abstract framework for instantiating CI, and also shows that a previously existing function family (PKC 2018) was already CI under the LWE assumption. In addition, our framework transparently generalizes to other settings, yielding new results: - We show how to instantiate certain forms of multi-input CI under the LWE assumption. Prior constructions either relied on a very strong ``brute-force-is-best\u27\u27 type of hardness assumption (Holmgren and Lombardi, FOCS 2018) or were restricted to ``output-only\u27\u27 relations (Zhandry, CRYPTO 2016). - We construct single-input CI hash functions from indistinguishability obfuscation (iO) and one-way permutations. Prior constructions relied essentially on variants of fully homomorphic encryption that are impossible to construct from such primitives. This result also generalizes to more expressive variants of multi-input CI under iO and additional standard assumptions

    Fiat-Shamir From Simpler Assumptions

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    We present two new protocols: (1) A succinct publicly verifiable non-interactive argument system for log-space uniform NC computations, under the assumption that any one of a broad class of fully homomorphic encryption (FHE) schemes has almost optimal security against polynomial-time adversaries. The class includes all FHE schemes in the literature that are based on the learning with errors (LWE) problem. (2) A non-interactive zero-knowledge argument system for NP in the common random string model, assuming almost optimal hardness of search-LWE against polynomial-time adversaries. Both results are obtained by applying the Fiat-Shamir transform with explicit, efficiently computable functions (specifically, correlation intractable functions) to certain classes of interactive proofs. We improve over prior work by reducing the security of these protocols to qualitatively weaker computational hardness assumptions. Along the way, we also show that the Fiat-Shamir transform can be soundly applied (in the plain model) to a richer class of protocols than was previously known

    Noninteractive Zero Knowledge for NP from (Plain) Learning With Errors

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    We finally close the long-standing problem of constructing a noninteractive zero-knowledge (NIZK) proof system for any NP language with security based on the plain Learning With Errors (LWE) problem, and thereby on worst-case lattice problems. Our proof system instantiates the framework recently developed by Canetti et al. [EUROCRYPT\u2718], Holmgren and Lombardi [FOCS\u2718], and Canetti et al. [STOC\u2719] for soundly applying the Fiat--Shamir transform using a hash function family that is correlation intractable for a suitable class of relations. Previously, such hash families were based either on ``exotic\u27\u27 assumptions (e.g., indistinguishability obfuscation or optimal hardness of certain LWE variants) or, more recently, on the existence of circularly secure fully homomorphic encryption (FHE). However, none of these assumptions are known to be implied by plain LWE or worst-case hardness. Our main technical contribution is a hash family that is correlation intractable for arbitrary size-SS circuits, for any polynomially bounded SS, based on plain LWE (with small polynomial approximation factors). The construction combines two novel ingredients: a correlation-intractable hash family for log-depth circuits based on LWE (or even the potentially harder Short Integer Solution problem), and a ``bootstrapping\u27\u27 transform that uses (leveled) FHE to promote correlation intractability for the FHE decryption circuit to arbitrary (bounded) circuits. Our construction can be instantiated in two possible ``modes,\u27\u27 yielding a NIZK that is either computationally sound and statistically zero knowledge in the common random string model, or vice-versa in the common reference string model

    Hiding secrets in public random functions

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    Constructing advanced cryptographic applications often requires the ability of privately embedding messages or functions in the code of a program. As an example, consider the task of building a searchable encryption scheme, which allows the users to search over the encrypted data and learn nothing other than the search result. Such a task is achievable if it is possible to embed the secret key of an encryption scheme into the code of a program that performs the "decrypt-then-search" functionality, and guarantee that the code hides everything except its functionality. This thesis studies two cryptographic primitives that facilitate the capability of hiding secrets in the program of random functions. 1. We first study the notion of a private constrained pseudorandom function (PCPRF). A PCPRF allows the PRF master secret key holder to derive a public constrained key that changes the functionality of the original key without revealing the constraint description. Such a notion closely captures the goal of privately embedding functions in the code of a random function. Our main contribution is in constructing single-key secure PCPRFs for NC^1 circuit constraints based on the learning with errors assumption. Single-key secure PCPRFs were known to support a wide range of cryptographic applications, such as private-key deniable encryption and watermarking. In addition, we build reusable garbled circuits from PCPRFs. 2. We then study how to construct cryptographic hash functions that satisfy strong random oracle-like properties. In particular, we focus on the notion of correlation intractability, which requires that given the description of a function, it should be hard to find an input-output pair that satisfies any sparse relations. Correlation intractability captures the security properties required for, e.g., the soundness of the Fiat-Shamir heuristic, where the Fiat-Shamir transformation is a practical method of building signature schemes from interactive proof protocols. However, correlation intractability was shown to be impossible to achieve for certain length parameters, and was widely considered to be unobtainable. Our contribution is in building correlation intractable functions from various cryptographic assumptions. The security analyses of the constructions use the techniques of secretly embedding constraints in the code of random functions

    IST Austria Thesis

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    A search problem lies in the complexity class FNP if a solution to the given instance of the problem can be verified efficiently. The complexity class TFNP consists of all search problems in FNP that are total in the sense that a solution is guaranteed to exist. TFNP contains a host of interesting problems from fields such as algorithmic game theory, computational topology, number theory and combinatorics. Since TFNP is a semantic class, it is unlikely to have a complete problem. Instead, one studies its syntactic subclasses which are defined based on the combinatorial principle used to argue totality. Of particular interest is the subclass PPAD, which contains important problems like computing Nash equilibrium for bimatrix games and computational counterparts of several fixed-point theorems as complete. In the thesis, we undertake the study of averagecase hardness of TFNP, and in particular its subclass PPAD. Almost nothing was known about average-case hardness of PPAD before a series of recent results showed how to achieve it using a cryptographic primitive called program obfuscation. However, it is currently not known how to construct program obfuscation from standard cryptographic assumptions. Therefore, it is desirable to relax the assumption under which average-case hardness of PPAD can be shown. In the thesis we take a step in this direction. First, we show that assuming the (average-case) hardness of a numbertheoretic problem related to factoring of integers, which we call Iterated-Squaring, PPAD is hard-on-average in the random-oracle model. Then we strengthen this result to show that the average-case hardness of PPAD reduces to the (adaptive) soundness of the Fiat-Shamir Transform, a well-known technique used to compile a public-coin interactive protocol into a non-interactive one. As a corollary, we obtain average-case hardness for PPAD in the random-oracle model assuming the worst-case hardness of #SAT. Moreover, the above results can all be strengthened to obtain average-case hardness for the class CLS ⊆ PPAD. Our main technical contribution is constructing incrementally-verifiable procedures for computing Iterated-Squaring and #SAT. By incrementally-verifiable, we mean that every intermediate state of the computation includes a proof of its correctness, and the proof can be updated and verified in polynomial time. Previous constructions of such procedures relied on strong, non-standard assumptions. Instead, we introduce a technique called recursive proof-merging to obtain the same from weaker assumptions

    Lossy Correlation Intractability and PPAD Hardness from Sub-exponential LWE

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    We introduce a new cryptographic primitive, a lossy correlation-intractable hash function, and use it to soundly instantiate the Fiat-Shamir transform for the general interactive sumcheck protocol, assuming sub-exponential hardness of the Learning with Errors (LWE) problem. By combining this with the result of Choudhuri et al. (STOC 2019), we show that #SAT\#\mathsf{SAT} reduces to end-of-line, which is a PPAD\mathsf{PPAD}-complete problem, assuming the sub-exponential hardness of LWE

    Fiat-Shamir and Correlation Intractability from Strong KDM-Secure Encryption

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    A hash function family is called correlation intractable if for all sparse relations, it is hard to find, given a random function from the family, an input-output pair that satisfies the relation (Canetti et al., STOC 98). Correlation intractability (CI) captures a strong Random-Oracle-like property of hash functions. In particular, when security holds for all sparse relations, CI suffices for guaranteeing the soundness of the Fiat-Shamir transformation from any constant round, statistically sound interactive proof to a non-interactive argument. However, to date, the only CI hash function for all sparse relations (Kalai et al., Crypto 17) is based on general program obfuscation with exponential hardness properties. We construct a simple CI hash function for arbitrary sparse relations, from any symmetric encryption scheme that satisfies some natural structural properties, and in addition guarantees that key recovery attacks mounted by polynomial-time adversaries have only exponentially small success probability - even in the context of key-dependent messages (KDM). We then provide parameter settings where ElGamal encryption and Regev encryption plausibly satisfy the needed properties. Our techniques are based on those of Kalai et al., with the main contribution being substituting a statistical argument for the use of obfuscation, therefore greatly simplifying the construction and basing security on better-understood intractability assumptions. In addition, we extend the definition of correlation intractability to handle moderately sparse relations so as to capture the properties required in proof-of-work applications (e.g. Bitcoin). We also discuss the applicability of our constructions and analyses in that regime

    Rational Modular Encoding in the DCR Setting: Non-Interactive Range Proofs and Paillier-Based Naor-Yung in the Standard Model

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    International audienceRange proofs allow a sender to convince a verifier that committed integers belong to an interval without revealing anything else. So far, all known non-interactive range proofs in the standard model rely on groups endowed with a bilinear map. Moreover, they either require the group order to be larger than the range of any proven statement or they suffer from a wasteful rate. Recently (Eurocrypt'21), Couteau et al. introduced a new approach to efficiently prove range membership by encoding integers as a modular ratio between small integers. We show that their technique can be transposed in the standard model under the Composite Residuosity (DCR) assumption. Interestingly, with this modification, the size of ranges is not a priori restricted by the common reference string. It also gives a constant ratio between the size of ranges and proofs. Moreover, we show that their technique of encoding messages as bounded rationals provides a secure standard model instantiation of the Naor-Yung CCA2 encryption paradigm under the DCR assumption

    Dual-Mode NIZKs from Obfuscation

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    Two standard security properties of a non-interactive zero-knowledge (NIZK) scheme are soundness and zero-knowledge. But while standard NIZK systems can only provide one of those properties against unbounded adversaries, dual-mode NIZK systems allow to choose dynamically and adaptively which of these properties holds unconditionally. The only known dual-mode NIZK systems are Groth-Sahai proofs (which have proved extremely useful in a variety of applications), and the FHE-based NIZK constructions of Canetti et al. and Peikert et al, which are concurrent and independent to this work. However, all these constructions rely on specific algebraic settings. Here, we provide a generic construction of dual-mode NIZK systems for all of NP. The public parameters of our scheme can be set up in one of two indistinguishable ways. One way provides unconditional soundness, while the other provides unconditional zero-knowledge. Our scheme relies on subexponentially secure indistinguishability obfuscation and subexponentially secure one-way functions, but otherwise only on comparatively mild and generic computational assumptions. These generic assumptions can be instantiated under any one of the DDH, k-LIN, DCR, or QR assumptions. As an application, we reduce the required assumptions necessary for several recent obfuscation-based constructions of multilinear maps. Combined with previous work, our scheme can be used to construct multilinear maps from obfuscation and a group in which the strong Diffie-Hellman assumption holds. We also believe that our work adds to the understanding of the construction of NIZK systems, as it provides a conceptually new way to achieve dual-mode properties
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