Advances in Cryptology – ASIACRYPT 2019: 25th International Conference on the Theory and Application of Cryptology and Information Security, Kobe, Japan, December 8–12, 2019, Proceedings, Part II

The Learning Parity with Noise (LPN) problem has recently found many cryptographic applications such as authentication protocols, pseudorandom generators/functions and even asymmetric tasks including public-key encryption (PKE) schemes and oblivious transfer (OT) protocols. It however remains a long-standing open problem whether LPN implies collision resistant hash (CRH) functions. Inspired by the recent work of Applebaum et al. (ITCS 2017), we introduce a general construction of CRH from LPN for various parameter choices. We show that, just to mention a few notable ones, under any of the following hardness assumptions (for the two most common variants of LPN) 1. constant-noise LPN is 2 0.5+ε -hard for any constant ε > 0; 2. constant-noise LPN is 2 log -hard given q = poly(n) samples; 3. low-noise LPN (of noise rate 1/ √ n) is 2 √ n/ log -hard given q = poly(n) samples. there exists CRH functions with constant (or even poly-logarithmic) shrinkage, which can be implemented using polynomial-size depth-3 circuits with NOT, (unbounded fan-in) AND and XOR gates. Our technical route LPN → bSVP → CRH is reminiscent of the known reductions for the large-modulus analogue, i.e., LWE → SIS → CRH, where the binary Shortest Vector Problem (bSVP) was recently introduced by Applebaum et al. (ITCS 2017) that enables CRH in a similar manner to Ajtai’s CRH functions based on the Short Integer Solution (SIS) problem. c © International Association for Cryptologic Research 2019 S. D. Galbraith and S. Moriai (Eds.): ASIACRYPT 2019, LNCS 11922, pp. 3–24, 2019. https://doi.org/10.1007/978-3-030-34621-8_1