Paper 2020/229

Tight Time-Space Lower Bounds for Finding Multiple Collision Pairs and Their Applications

Itai Dinur

Abstract

We consider a collision search problem (CSP), where given a parameter $C$, the goal is to find $C$ collision pairs in a random function $f:[N]\to [N]$ (where $[N]=\{0,1,\dots ,N-1\})$ using $S$ bits of memory. Algorithms for CSP have numerous cryptanalytic applications such as space-efficient attacks on double and triple encryption. The best known algorithm for CSP is parallel collision search (PCS) published by van Oorschot and Wiener, which achieves the time-space tradeoff $T^2\cdot S=\tilde{O}(C^2\cdot N)$ for $S=\tilde{O}(C)$. In this paper, we prove that any algorithm for CSP satisfies $T^2\cdot S=\tilde{\mathrm{\Omega }}(C^2\cdot N)$ for $S=\tilde{O}(C)$, hence the best known time-space tradeoff is optimal (up to poly-logarithmic factors in $N$). On the other hand, we give strong evidence that proving similar unconditional time-space tradeoff lower bounds on CSP applications (such as breaking double and triple encryption) may be very difficult, and would imply a breakthrough in complexity theory. Hence, we propose a new restricted model of computation and prove that under this model, the best known time-space tradeoff attack on double encryption is optimal.

Note: Minor changes