Solving the LPN problem in cube-root time

TL;DR

This paper presents a novel algorithm that solves the Learning from Parity with Noise (LPN) problem in approximately cube root time, significantly improving over brute-force methods by leveraging techniques from fast correlation attacks.

Contribution

It introduces a new approach to solve LPN in near cube root time using known correlation attack techniques, with explicit bounds on the number of equations needed for efficiency.

Findings

Algorithm reduces LPN solving time to near cube root under certain conditions.

Performance depends on the number of equations and bias of noisy equations.

Complexity can be further reduced when bounds are exceeded.

Abstract

In this paper it is shown that given a sufficient number of (noisy) random binary linear equations, the Learning from Parity with Noise (LPN) problem can be solved in essentially cube root time in the number of unknowns. The techniques used to recover the solution are known from fast correlation attacks on stream ciphers. As in fast correlation attacks, the performance of the algorithm depends on the number of equations given. It is shown that if this number exceeds a certain bound, and the bias of the noisy equations is polynomial in number of unknowns, the running time of the algorithm is reduced to almost cube root time compared to the brute force checking of all possible solutions. The mentioned bound is explicitly given and it is further shown that when this bound is exceeded, the complexity of the approach can even be further reduced.