International Association for Cryptologic Research

International Association
for Cryptologic Research


Fermi Ma


New Techniques for Obfuscating Conjunctions 📺
A conjunction is a function $$f(x_1,\dots ,x_n) = \bigwedge _{i \in S} l_i$$ where $$S \subseteq [n]$$ and each $$l_i$$ is $$x_i$$ or $$\lnot x_i$$. Bishop et al. (CRYPTO 2018) recently proposed obfuscating conjunctions by embedding them in the error positions of a noisy Reed-Solomon codeword and placing the codeword in a group exponent. They prove distributional virtual black box (VBB) security in the generic group model for random conjunctions where $$|S| \ge 0.226n$$. While conjunction obfuscation is known from LWE [31, 47], these constructions rely on substantial technical machinery.In this work, we conduct an extensive study of simple conjunction obfuscation techniques. We abstract the Bishop et al. scheme to obtain an equivalent yet more efficient “dual” scheme that can handle conjunctions over exponential size alphabets. This scheme admits a straightforward proof of generic group security, which we combine with a novel combinatorial argument to obtain distributional VBB security for |S| of any size.If we replace the Reed-Solomon code with a random binary linear code, we can prove security from standard LPN and avoid encoding in a group. This addresses an open problem posed by Bishop et al. to prove security of this simple approach in the standard model.We give a new construction that achieves information theoretic distributional VBB security and weak functionality preservation for $$|S| \ge n - n^\delta $$ and $$\delta < 1$$. Assuming discrete log and $$\delta < 1/2$$, we satisfy a stronger notion of functionality preservation for computationally bounded adversaries while still achieving information theoretic security.
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).
On the (In)security of Kilian-Based SNARGs
The Fiat-Shamir transform is an incredibly powerful technique that uses a suitable hash function to reduce the interaction of general public-coin protocols. Unfortunately, there are known counterexamples showing that this methodology may not be sound (no matter what concrete hash function is used). Still, these counterexamples are somewhat unsatisfying, as the underlying protocols were specifically tailored to make Fiat-Shamir fail. This raises the question of whether this transform is sound when applied to natural protocols.One of the most important protocols for which we would like to reduce interaction is Kilian’s four-message argument system for all of $$\mathsf {NP}$$ , based on collision resistant hash functions ( $$\mathsf {CRHF}$$ ) and probabilistically checkable proofs ( $$\mathsf {PCP}$$ s). Indeed, an application of the Fiat-Shamir transform to Kilian’s protocol is at the heart of both theoretical results (e.g., Micali’s CS proofs) as well as leading practical approaches of highly efficient non-interactive proof-systems (e.g., $$\mathsf {SNARK}$$ s and $$\mathsf {STARK}$$ s).In this work, we show significant obstacles to establishing soundness of (what we refer to as) the “Fiat-Shamir-Kilian-Micali” ( $$\mathsf {FSKM}$$ ) protocol. More specifically:We construct a (contrived) $$\mathsf {CRHF}$$ for which $$\mathsf {FSKM}$$ is unsound for a very large class of $$\mathsf {PCP}$$ s and for any Fiat-Shamir hash function. The collision-resistance of our $$\mathsf {CRHF}$$ relies on very strong but plausible cryptographic assumptions. The statement is “tight” in the following sense: any $$\mathsf {PCP}$$ outside the scope of our result trivially implies a $$\mathsf {SNARK}$$ , eliminating the need for $$\mathsf {FSKM}$$ in the first place.Second, we consider a known extension of Kilian’s protocol to an interactive variant of $$\mathsf {PCP}$$ s called probabilistically checkable interactive proofs ( $$\mathsf {PCIP})$$ (also known as interactive oracle proofs or $$\mathsf {IOP}$$ s). We construct a particular (contrived) $$\mathsf {PCIP}$$ for $$\mathsf {NP}$$ for which the $$\mathsf {FSKM}$$ protocol is unsound no matter what $$\mathsf {CRHF}$$ and Fiat-Shamir hash function is used. This result is unconditional (i.e., does not rely on any cryptographic assumptions). Put together, our results show that the soundness of $$\mathsf {FSKM}$$ must rely on some special structure of both the $$\mathsf {CRHF}$$ and $$\mathsf {PCP}$$ that underlie Kilian’s protocol. We believe these negative results may cast light on how to securely instantiate the $$\mathsf {FSKM}$$ protocol by a synergistic choice of the $$\mathsf {PCP}$$ , $$\mathsf {CRHF}$$ , and Fiat-Shamir hash function.
Public-Key Function-Private Hidden Vector Encryption (and More)
We construct public-key function-private predicate encryption for the “small superset functionality,” recently introduced by Beullens and Wee (PKC 2019). This functionality captures several important classes of predicates:Point functions. For point function predicates, our construction is equivalent to public-key function-private anonymous identity-based encryption.Conjunctions. If the predicate computes a conjunction, our construction is a public-key function-private hidden vector encryption scheme. This addresses an open problem posed by Boneh, Raghunathan, and Segev (ASIACRYPT 2013).d-CNFs and read-once conjunctions of d-disjunctions for constant-size d. Our construction extends the group-based obfuscation schemes of Bishop et al. (CRYPTO 2018), Beullens and Wee (PKC 2019), and Bartusek et al. (EUROCRYPT 2019) to the setting of public-key function-private predicate encryption. We achieve an average-case notion of function privacy, which guarantees that a decryption key $$\mathsf {sk} _f$$ reveals nothing about f as long as f is drawn from a distribution with sufficient entropy. We formalize this security notion as a generalization of the (enhanced) real-or-random function privacy definition of Boneh, Raghunathan, and Segev (CRYPTO 2013). Our construction relies on bilinear groups, and we prove security in the generic bilinear group model.
The MMap Strikes Back: Obfuscation and New Multilinear Maps Immune to CLT13 Zeroizing Attacks
Fermi Ma Mark Zhandry
All known multilinear map candidates have suffered from a class of attacks known as “zeroizing” attacks, which render them unusable for many applications. We provide a new construction of polynomial-degree multilinear maps and show that our scheme is provably immune to zeroizing attacks under a strengthening of the Branching Program Un-Annihilatability Assumption (Garg et al., TCC 2016-B).Concretely, we build our scheme on top of the CLT13 multilinear maps (Coron et al., CRYPTO 2013). In order to justify the security of our new scheme, we devise a weak multilinear map model for CLT13 that captures zeroizing attacks and generalizations, reflecting all known classical polynomial-time attacks on CLT13. In our model, we show that our new multilinear map scheme achieves ideal security, meaning no known attacks apply to our scheme. Using our scheme, we give a new multiparty key agreement protocol that is several orders of magnitude more efficient that what was previously possible.We also demonstrate the general applicability of our model by showing that several existing obfuscation and order-revealing encryption schemes, when instantiated with CLT13 maps, are secure against known attacks. These are schemes that are actually being implemented for experimentation, but until our work had no rigorous justification for security.
Return of GGH15: Provable Security Against Zeroizing Attacks
The GGH15 multilinear maps have served as the foundation for a number of cutting-edge cryptographic proposals. Unfortunately, many schemes built on GGH15 have been explicitly broken by so-called “zeroizing attacks,” which exploit leakage from honest zero-test queries. The precise settings in which zeroizing attacks are possible have remained unclear. Most notably, none of the current indistinguishability obfuscation (iO) candidates from GGH15 have any formal security guarantees against zeroizing attacks.In this work, we demonstrate that all known zeroizing attacks on GGH15 implicitly construct algebraic relations between the results of zero-testing and the encoded plaintext elements. We then propose a “GGH15 zeroizing model” as a new general framework which greatly generalizes known attacks.Our second contribution is to describe a new GGH15 variant, which we formally analyze in our GGH15 zeroizing model. We then construct a new iO candidate using our multilinear map, which we prove secure in the GGH15 zeroizing model. This implies resistance to all known zeroizing strategies. The proof relies on the Branching Program Un-Annihilatability (BPUA) Assumption of Garg et al. [TCC 16-B] (which is implied by PRFs in $$\mathsf {NC}^1$$ secure against $$\mathsf {P}/\mathsf {poly}$$) and the complexity-theoretic p-Bounded Speedup Hypothesis of Miles et al. [ePrint 14] (a strengthening of the Exponential Time Hypothesis).