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Title:
Cryptographic Hashing From Strong One-Way Functions
Authors: Justin Holmgren, Alex Lombardi
Abstract:Constructing collision-resistant hash families (CRHFs) from one-way functions is a long-standing open problem and source of frustration in theoretical cryptography. In fact, there are strong negative results: black-box separations from one-way functions that are 2^{-(1-o(1))n}-secure against polynomial time adversaries (Simon, EUROCRYPT ‘98) and even from indistinguishability obfuscation (Asharov and Segev, FOCS ‘15). In this work, we formulate a mild strengthening of exponentially secure one-way functions, and we construct CRHFs from such functions. Specifically, our security notion requires that every polynomial time algorithm has at most 2^{-n - \omega(\log(n))} probability of inverting two independent challenges. More generally, we consider the problem of simultaneously inverting k functions f_1,\ldots, f_k, which we say constitute a ``one-way product function’’ (OWPF). We show that sufficiently hard OWPFs yield hash families that are multi-input correlation intractable (Canetti, Goldreich, and Halevi, STOC '98) with respect to all sparse (bounded arity) output relations. Additionally assuming indistinguishability obfuscation, we construct hash families that achieve a broader notion of correlation intractability, extending the recent work of Kalai, Rothblum, and Rothblum (CRYPTO '17). In particular, these families are sufficient to instantiate the Fiat-Shamir heuristic in the plain model for a natural class of interactive proofs. An interesting consequence of our results is a potential new avenue for bypassing black-box separations. In particular, proving (with necessarily non-black-box techniques) that parallel repetition amplifies the hardness of specific one-way functions – for example, all one-way permutations – suffices to directly bypass Simon’s impossibility result.
ePrint: https://eprint.iacr.org/2018/385
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