Using Random Error Correcting Codes in Near-Collision Attacks on Generic Hash-Functions

Abstract

In this paper we consider the problem of finding near- collisions with Hamming distance bounded by $r$ in generic $n$-bit hash functions. In 2011, Lamberger and Rijmen proposed a modified version of Pollard’s rho method, and in 2012 Leurent improved this memoryless algorithm by using any available memory to store chain endpoints. Both algorithms use a perfect error correcting code to change near-collisions into full-collisions, but such codes are rare and have very small distance. In this paper we propose using randomly chosen linear codes, whose decoding can be made efficient by using some of the available memory to store error-correction tables. Compared to Leurent’s algorithm, we experimentally verified an improvement ratio of about $3$ in a small example with $n=160$ and $r=33$ which we implemented on a single PC, and mathematically predicted a significant improvement ratio of about $730$ in a larger example with $n=1024$ and $r=100$, using $2^{40}$ memory.