Collision Resistance from Multi-collision Resistance

Abstract

Collision-resistant hash functions ( \(\textsf{CRH}\) ) are a fundamental and ubiquitous cryptographic primitive. Several recent works have studied a relaxation of \(\textsf{CRH}\) called t-way multi-collision-resistant hash functions ( \(t\text {-}\textsf{MCRH}\) ). These are families of functions for which it is computationally hard to find a t-way collision, even though such collisions are abundant (and even \((t-1)\) -way collisions may be easy to find). The case of \(t=2\) corresponds to standard \(\textsf{CRH}\) , but it is natural to study t- \(\textsf{MCRH}\) for larger values of t. Multi-collision resistance seems to be a qualitatively weaker property than standard collision resistance. Nevertheless, in this work we show a non-blackbox transformation of any moderately shrinking t- \(\textsf{MCRH}\) , for \(t \in \{3,4\}\) , into an (infinitely often secure) \(\textsf{CRH}\) . This transformation is non-constructive—we can prove the existence of a \(\textsf{CRH}\) but cannot explicitly point out a construction. Our result partially extends to larger values of t. In particular, we show that for suitable values of \(t>t'\) , we can transform a t- \(\textsf{MCRH}\) into a \(t'\) - \(\textsf{MCRH}\) , at the cost of reducing the shrinkage of the resulting hash function family and settling for infinitely often security. This result utilizes the list-decodability properties of Reed–Solomon codes.