Collision Resistance from Multi-collision Resistance for All Constant Parameters

A $t$-multi-collision-resistant hash function ($t$-MCRH) is a family of shrinking functions for which it is computationally hard to find $t$ distinct inputs mapping to the same output for a function sampled from this family. Several works have shown that $t$-MCRHs are sufficient for many of the applications of collision-resistant hash functions (CRHs), which correspond to the special case of $t = 2$.

An important question is hence whether $t$-MCRHs for $t > 2$ are fundamentally weaker objects than CRHs. As a first step towards resolving this question, Rothblum and Vasudevan (CRYPTO '22) recently gave non-black-box constructions of infinitely-often secure CRHs from $t$-MCRHs for $t \in \{3,4\}$ assuming the MCRH is sufficiently shrinking. Earlier on, Komargodski and Yogev (CRYPTO '18) also showed that $t$-MCRHs for any constant $t$ imply the weaker notion of a distributional CRH.

In this paper, we remove the limitations of prior works, and completely resolve the question of the power of $t$-MCRHs for constant $t$ in the infinitely-often regime, showing that the existence of such a function family always implies the existence of an infinitely-often secure CRH. As in the works mentioned above, our construction is non-blackbox and non-constructive. We further give a new domain extension result for MCRHs that enables us to show that the underlying MCRH need only have arbitrarily small linear shrinkage (mapping $(1 + \epsilon)n$ bits to $n$ bits for any fixed $\epsilon > 0$) to imply the existence of CRHs.