Links between Quantum Distinguishers Based on Simon’s Algorithm and Truncated Differentials

Abstract

In this paper, we study the quantum security of block ciphers based on Simon’s period-finding quantum algorithm. We explored the relations between periodic functions and truncated differentials. The basic observation is that truncated differentials with a probability of 1 can be used to construct periodic functions, and two such constructions are presented with the help of a new notion called difference-annihilation matrix. This technique releases us from the tedious manual work of verifying the period of functions. Based on these new constructions, we find an 8-round quantum distinguisher for LBlock and a 9/10/11/13/15-round quantum distinguisher for SIMON-32/48/64/96/128 which are the best results as far as we know. Besides, to explore the security bounds of block cipher structures against Simon’s algorithm based quantum attacks, the unified structure, which unifies the Feistel, Lai-Massey, and most generalized Feistel structures, is studied. We estimate the exact round number of probability 1 truncated differentials that one can construct. Based on these results, one can easily check the quantum security of specific block ciphers that are special cases of unified structures, when the details of the non-linear building blocks are not considered.