International Association
for Cryptologic Research

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.

Deliberately weakened ciphers are of great interest in political discussion on law enforcement, as in the constantly recurring crypto wars, and have been put in the spotlight of academics by recent progress. A paper at Eurocrypt 2021 showed a strong indication that the security of the widely-deployed stream cipher GEA-1 was deliberately and secretly weakened to 40 bits in order to fulfill European export restrictions that have been in place in the late 1990s. However, no explanation of how this could have been constructed was given. On the other hand, we have seen the MALICIOUS design framework, published at CRYPTO 2020, that allows to construct tweakable block ciphers with a backdoor, where the difficulty of recovering the backdoor relies on well-understood cryptographic assumptions. The constructed tweakable block cipher however is rather unusual and very different from, say, general-purpose ciphers like the AES. In this paper, we pick up both topics. For GEA-1 we thoroughly explain how the weakness was constructed, solving the main open question of the work mentioned above. By generalizing MALICIOUS we - for the first time - construct backdoored tweakable block ciphers that follow modern design principles for general-purpose block ciphers, i.e., more natural-looking deliberately weakened tweakable block ciphers.

There are many recent results on reverse-engineering (potentially hidden) structure in cryptographic S-boxes. The problem of recovering structure in the other main building block of symmetric cryptographic primitives, namely, the linear layer, has not been paid that much attention so far. To fill this gap, in this work, we develop a systematic approach to decomposing structure in the linear layer of a substitutionpermutation network (SPN), covering the case in which the specification of the linear layer is obfuscated by applying secret linear transformations to the S-boxes. We first present algorithms to decide whether an ms x ms matrix with entries in a prime field Fp can be represented as an m x m matrix over the extension field Fps . We then study the case of recovering structure in MDS matrices by investigating whether a given MDS matrix follows a Cauchy construction. As an application, for the first time, we show that the 8 x 8 MDS matrix over F28 used in the hash function Streebog is a Cauchy matrix.

In this paper, we study the effect of two modifications to multivariate public key encryption schemes: internal perturbation (ip), and Q_+. Focusing on the Dob encryption scheme, a construction utilising these modifications, we accurately predict the number of degree fall polynomials produced in a Gröbner basis attack, up to and including degree five. The predictions remain accurate even when fixing variables. Based on this new theory we design a novel attack on the Dob encryption scheme, which breaks Dob using the parameters suggested by its designers. While our work primarily focuses on the Dob encryption scheme, we also believe that the presented techniques will be of particular interest to the analysis of other big-field schemes.