Faster Algorithms for the Sparse Random 3XOR Problem

Algorithms for the Sparse Random 3XOR Problem

Charles Bouillaguet (1) , Claire Delaplace (2)

1 CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
2 Horst Görtz Institute for IT-security

We present two new algorithms for a variant of the 3XOR problem with lists consisting of N n-bit 10 vectors whose coefficients are drawn randomly according to a Bernoulli distribution of parameter 11 p < 1/2. We show that in this particular context the problem can be solved much more efficiently 12 than in the general setting. In particular, we present two new algorithms. The first one has a 13 time complexity which is both O N 1+2.583p and O N 2−(1−2p) 2.1. The second one has a time 14 complexity which is almost linear in N for small values of p p ≤ 0.15 and has a time complexity of 15Õ N 2−1.97(1−2p) 2.37 for p > 0.13. The analysis of these algorithms reveal a "phase change" for a 16 certain threshold p. 17 2012 ACM Subject Classification Theory of computation → Computational complexity and cryp-18 tography; Theory of computation 19

Mots clés: 3-xor problem; and phrases Algorithms; random sparse 3-xor

Domaines: Cryptographie et sécurité [cs.CR]

Charles Bouillaguet, Claire Delaplace. Algorithms for the Sparse Random 3XOR Problem. 2020. ⟨hal-02306917v2⟩