CryptoDB
Lossy Cryptography from Code-Based Assumptions
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| Conference: | CRYPTO 2024 |
| Abstract: | 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 $\mathcal{SZK}$ (statistical zero-knowledge); as a consequence, they can only be based on assumptions that are broken in $\mathcal{BPP}^{\mathcal{SZK}}$. This poses a barrier for building advanced cryptography from code-based assumptions such as Learning Parity with Noise (LPN), as LPN is only known to be in $\mathcal{BPP}^{\mathcal{SZK}}$ under an extremely low noise rate $\frac{\log^2 n}{n}$, for which it 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 $\mathcal{BPP}^{\mathcal{SZK}}$ 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 $k\mbox{-}$XOR in average-case complexity. Roughly, the assumption states that \[(\mathbf{T}\, \mathbf{M}, \mathbf{s} \,\mathbf{T}\, \mathbf{M} + \mathbf{e}) \quad \text{is indistinguishable from}\quad (\mathbf{T} \,\mathbf{M}, \mathbf{u}),\] for a random (dense) matrix $\mathbf{T}$, random sparse matrix $\mathbf{M}$, and sparse noise vector $\mathbf{e}$ 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 $\frac{\log^2 n}{n}$ that is only quasi-polynomially secure. |
BibTeX
@inproceedings{crypto-2024-34277,
title={Lossy Cryptography from Code-Based Assumptions},
publisher={Springer-Verlag},
author={Quang Dao and Aayush Jain},
year=2024
}