Paper 2003/065

Hash Function Balance and its Impact on Birthday Attacks

Mihir Bellare and Tadayoshi Kohno

Abstract

Textbooks tell us that a birthday attack on a hash function $h$ with range size $r$ requires $r^{1/2}$ trials (hash computations) to find a collision. But this is misleading, being true only if $h$ is regular, meaning all points in the range have the same number of pre-images under $h$; if $h$ is not regular, \textit{fewer} trials may be required. But how much fewer? This paper addresses this question by introducing a measure of the ``amount of regularity'' of a hash function that we call its balance, and then providing estimates of the success-rate of the birthday attack as a function of the balance of the hash function being attacked. In particular, we will see that the number of trials to find a collision can be significantly less than $r^{1/2}$ for hash functions of low balance. This leads us to examine popular design principles, such as the MD (Merkle-Damgård) transform, from the point of view of balance preservation, and to mount experiments to determine the balance of popular hash functions.