Paper 2009/525

On Quantifying the Resistance of Concrete Hash Functions to Generic Multi-Collision Attacks

Somindu C. Ramanna and Palash Sarkar

Abstract

Bellare and Kohno (2004) introduced the notion of balance to quantify the resistance of a hash function $h$ to a generic collision attack. Motivated by their work, we consider the problem of quantifying the resistance of $h$ to a generic multi-collision attack. To this end, we introduce the notion of $r$-balance $\mu_r (h)$ of $h$ and obtain bounds on the success probability of finding an $r$-collision in terms of $\mu_r (h)$. These bounds show that for a hash function with $m$ image points, if the number of trials $q$ is $\Theta(rm^{(r-1)\mu_r(h)/r})$, then it is possible to find $r$-collisions with a significant probability of success. The behaviour of random functions and the expected number of trials to obtain an $r$-collision is studied. These results extend and complete the earlier results obtained by Bellare and Kohno (2004) for collisions (i.e., $r = 2$). Going beyond their work, we provide a new design criteria to provide quantifiable resistance to generic multi- collision attacks. Further, we make a detailed probabilistic investigation of the variation of $r$-balance over the set of all functions and obtain support for the view that most functions have $r$-balance close to one.