On the Wrong Key Randomisation and Key Equivalence Hypotheses in Matsui’s Algorithm 2

This paper aims to improve the understanding of the complexities for Matsui’s Algorithm 2 — one of the most well-studied and powerful cryptanalytic techniques available for block ciphers today. We start with the observation that the standard interpretation of the wrong key randomisation hypothesis needs adjustment. We show that it systematically neglects the varying bias for wrong keys. Based on that, we propose an adjusted statistical model and derive more accurate estimates for the success probability and data complexity of linear attacks which are demonstrated to deviate from all known estimates. Our study suggests that the efficiency of Matsui’s Algorithm 2 has been previously somewhat overestimated in the cases where the adversary attempts to use a linear approximation with a low bias, to attain a high computational advantage over brute force, or both. These cases are typical since cryptanalysts always try to break as many rounds of the cipher as possible by pushing the attack to its limit. Surprisingly, our approach also reveals the fact that the success probability is not a monotonously increasing function of the data complexity, and can decrease if more data is used. Using less data can therefore result in a more powerful attack. A second assumption usually made in linear cryptanalysis is the key equivalence hypothesis, even though due to the linear hull effect, the bias can heavily depend on the key. As a further contribution of this paper, we propose a practical technique that aims to take this into account. All theoretical observations and techniques are accompanied by experiments with small-scale ciphers.