Provably good codes for hash function design

Abstract

We develop a new technique to lower bound the minimum distance of quasi-cyclic codes with large dimension by reducing the problem to lower bounding the minimum distance of a few significantly smaller dimensional codes. Using this technique, we prove that a code which is similar to the SHA-1 message expansion code has minimum distance at least 82, and that too in just the last 64 of the 80 expanded words. Further the minimum weight in the last 60 words (last 48 words) is at least 75 (52 respectively). We expect our technique to be helpful in designing future practical collision-resistant hash functions. We also use the technique to find the minimum weight of the SHA-1 code (25 in the last 60 words), which was an open problem. © Springer-Verlag Berlin Heidelberg 2007.