Lossy Cryptography from Code-Based Assumptions

Over the past few decades, we have seen a proliferation of advanced cryptographic primitives with lossy or homomorphic properties built from various assumptions such as Quadratic Residuosity, Decisional Diffie-Hellman, and Learning with Errors. These primitives imply hard problems in the complexity class SZK (statistical zero-knowledge); as a consequence, they can only be based on assumptions that are broken in BPP SZK . This poses a barrier for building advanced primitives from code-based assumptions, as the only known such assumption is Learning Parity with Noise (LPN) with an extremely low noise rate log 2 n n , which is broken in quasi-polynomial time.

In this work, we propose a new code-based assumption: Dense-Sparse LPN, that falls in the complexity class BPP SZK and is conjectured to be secure against subexponential time adversaries. Our assumption is a variant of LPN that is inspired by McEliece's cryptosystem and random k-XOR in average-case complexity. Roughly, the assumption states that (T M, s T M + e) is indistinguishable from (T M, u), for a random (dense) matrix T, random sparse matrix M, and sparse noise vector e drawn from the Bernoulli distribution with inverse polynomial noise probability.

We leverage our assumption to build lossy trapdoor functions (Peikert-Waters STOC 08). This gives the first post-quantum alternative to the lattice-based construction in the original paper. Lossy trapdoor functions, being a fundamental cryptographic tool, are known to enable a broad spectrum of both lossy and non-lossy cryptographic primitives; our construction thus implies these primitives in a generic manner. In particular, we achieve collision-resistant hash functions with plausible subexponential security, improving over a prior construction from LPN with noise rate log 2 n n that is only quasi-polynomially secure.