On the Hardness of Learning Parity with Noise over Rings

Abstract

Learning Parity with Noise (LPN) represents a notoriously hard problem in learning theory and it is also closely related to the “decoding random linear codes” problem in coding theory. Recently LPN has found many cryptographic applications such as authentication protocols, pseudorandom generators/functions and even advanced tasks including public-key encryption (PKE) schemes and oblivious transfer (OT) protocols. Crypto-systems based on LPN are computationally efficient and parallelizable in concept, thanks to the simple algebraic structure of LPN, but they (especially the public-key ones) are typically inefficient in terms of public-key/ciphertext sizes and/or communication complexity. To mitigate the issue, Heyse et al. (FSE 2012) introduced the ring variant of LPN (Ring-LPN) that enjoys a compact structure and gives rise to significantly more efficient cryptographic schemes. However, unlike its large-modulus analogue Ring-LWE (to which a reduction from ideal lattice problems can be established), no formal asymptotic studies are known for the security of Ring-LPN or its connections to other hardness assumptions.