Cryptographic Hashing from Strong One-Way Functions (Or: One-Way Product Functions and Their Applications)

• Published in: 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) pp. 850 - 858
• Main Authors: Holmgren, Justin, Lombardi, Alex
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.10.2018
• Subjects: Buildings; Correlation; hardness amplification; hash functions; multi instance security; random oracles; Resistance
• ISSN: 2575-8454
• DOI: 10.1109/FOCS.2018.00085

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 exponentially 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 an extremely small probability of inverting two independent challenges. More generally, we consider the problem of simultaneously inverting k functions, 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.