International Association
for Cryptologic Research

We put forth a new primitive called obliviously programmable function (OPF) to construct two random-oracle-like primitives: – Universal computational extractors (UCEs), introduced by Bellare, Hoang, and Keelveedhi [3], can securely replace random oracles in various applications, including KDM-secure encryption, deterministic encryption, RSA-OAEP, universal hardcore bits, etc. – Multi-bit point obfuscation with auxiliary input (MB-AIPO). It enables upgrading CPA-secure public-key encryption (PKE) into a CCA-secure one [30] and serves as a tool to instantiate the ran- dom oracles used in the Fujisaki-Okamoto transform for lossy PKEs [32]. Despite their usefulness, constructing UCEs and MB-AIPO in the standard model is challenging. The existing constructions of both primitives [15,16] use indistinguishability obfuscation (iO) plus point functions with auxiliary input (AIPO). OPF can replace the use iO in the constructions of UCE and MB-AIPO. We use OPF plus AIPO to construct – UCE with one query for strongly unpredictable sources, – MB-AIPO for strongly unpredictable distributions and – PKE scheme that is IND-CPA secure in the presence of computationally uninvertible leakage on the secret key. We then construct OPF for NC1 circuits from lattice assumptions based on the GGH15 encodings [23], without using iO. In sum, we give new constructions of the above three primitives under the following assumptions: (1) LWE with subexponential hardness; (2) private-coin evasive LWE as- sumption for specific samplers; (3) the existence of AIPO in NC1. As a byproduct, we construct an ‘NC1-universal AIPO’ under the assumptions (1) and (2).

Two of the most useful cryptographic primitives that can be constructed from one-way functions are pseudorandom generators (PRGs) and universal one-way hash functions (UOWHFs). The three major efficiency measures of these primitives are: seed length, number of calls to the one-way function, and adaptivity of these calls. Although a long and successful line of research studied these primitives, their optimal efficiency is not yet fully understood: there are gaps between the known upper bounds and the known lower bounds for black-box constructions. Interestingly, the first construction of PRGs by H ̊astad, Impagliazzo, Levin, and Luby [SICOMP ’99], and the UOWHFs construction by Rompel [STOC ’90] shared a similar structure. Since then, there was an improvement in the efficiency of both constructions: The state-of-the-art construction of PRGs by Haitner, Reingold, and Vadhan [STOC ’10] uses O(n^4) bits of random seed and O(n^3) non-adaptive calls to the one-way function, or alternatively, seed of size O(n^3) with O(n^3) adaptive calls (Vadhan and Zhen [STOC ’12]). Constructing a UOWHF with similar parameters is still an open question. Currently, the best UOWHF construction by Haitner, Holenstein, Reingold, Vadhan, and Wee [Eurocrypt ’10] uses O(n^13) adaptive calls and a key of size O(n^5). In this work we give the first non-adaptive construction of UOWHFs from arbitrary one-way functions. Our construction uses O(n^9) calls to the one-way function, and a key of length O(n^10). By the result of Applebaum, Ishai, and Kushilevitz [FOCS ’04], the above implies the existence of UOWHFs in NC0, given the existence of one-way functions in NC1. We also show that the PRG construction of Haitner et al., with small modifications, yields a relaxed notion of UOWHFs. In order to analyze this construction, we introduce the notion of next-bit unreachable entropy, which replaces the next-bit pseudoentropy notion, used in the PRG construction above.

Constructing key-agreement protocols in the random oracle model (ROM) is a viable method to assess the feasibility of developing public-key cryptography within Minicrypt. Unfortunately, as shown by Impagliazzo and Rudich (STOC 1989) and Barak and Mahmoody (Crypto 2009), such protocols can only guarantee limited security: any $\ell$-query protocol can be attacked by an $O(\ell^2)$-query adversary. This quadratic gap matches the key-agreement protocol proposed by Merkle (CACM 78), known as Merkle's Puzzles. Besides query complexity, the communication complexity of key-agreement protocols in the ROM is also an interesting question in the realm of find-grained cryptography, even though only limited security is achievable. Haitner et al. (ITCS 2019) first observed that in Merkle's Puzzles, to obtain secrecy against an eavesdropper with $O(\ell^2)$ queries, the honest parties must exchange $\Omega(\ell)$ bits. Therefore, they conjectured that high communication complexity is unavoidable, any $\ell$-query protocols with $c$ bits of communication could be attacked by an $O(c\cdot \ell)$-query adversary. This, if true, will suggest that Merkle's Puzzle is also optimal regarding communication complexity. Building upon techniques from communication complexity, Haitner et al. (ITCS 2019) confirmed this conjecture for two types of key agreement protocols with certain natural properties. This work affirms the above conjecture for all non-adaptive protocols with perfect completeness. Our proof uses a novel idea called \textit{density increment argument}. This method could be of independent interest as it differs from previous communication lower bounds techniques (and bypasses some technical barriers).