International Association
for Cryptologic Research

In this paper we, for the first time, study the question under which circumstances decomposing a round function of a Substitution-Permutation Network is possible uniquely. More precisely, we provide necessary and sufficient criteria for the non-linear layer on when a decomposition is unique. Our results in particular imply that, when cryptographically strong S-boxes are used, the decomposition is indeed unique. We then apply our findings to the notion of alignment, pointing out that the previous definition allows for primitives that are both aligned and unaligned simultaneously. As a second result, we present experimental data that shows that alignment might only have limited impact. For this, we compare aligned and unaligned versions of the cipher PRESENT.

Recent constructions of (tweakable) block ciphers with an embedded cryptographic backdoor relied on the existence of probability-one differentials or perfect (non-)linear approximations over a reduced-round version of the primitive. In this work, we study how the existence of probability-one differentials or perfect linear approximations over two rounds of a substitution permutation network can be avoided by design. More precisely, we develop criteria on the s-box and the linear layer that guarantee the absence of probability-one differentials for all keys. We further present an algorithm that allows to efficiently exclude the existence of keys for which there exists a perfect linear approximation.

About 20 years ago, Wagner showed that most of the (then) known techniques used in the cryptanalysis of block ciphers were particular cases of what he called commutative diagram cryptanalysis. However, to the best of our knowledge, this general framework has not yet been leveraged to find concrete attacks.In this paper, we focus on a particular case of this framework and develop commutative cryptanalysis, whereby an attacker targeting a primitive E constructs affine permutations A and B such that E ○ A = B ○ E with a high probability, possibly for some weak keys. We develop the tools needed for the practical use of this technique: first, we generalize differential uniformity into “A-uniformity” and differential trails into “commutative trails”, and second we investigate the commutative behaviour of S-box layers, matrix multiplications, and key additions.Equipped with these new techniques, we find probability-one distinguishers using only two chosen plaintexts for large classes of weak keys in both a modified Midori and in Scream. For the same weak keys, we deduce high probability truncated differentials that can cover an arbitrary number of rounds, but which do not correspond to any high probability differential trails. Similarly, we show the existence of a trade-off in our variant of Midori whereby the probability of the commutative trail can be decreased in order to increase the weak key density. We also show some statistical patterns in the AES super S-box that have a much higher probability than the best differentials, and which hold for a class of weak keys of density about 2−4.5.

We give the first practical instance – bison – of the Whitened Swap-Or-Not construction. After clarifying inherent limitations of the construction, we point out that this way of building block ciphers allows easy and very strong arguments against differential attacks.