Automated cryptanalysis has taken center stage in the arena of cryptanalysis since the pioneering work by Mouhaet al. which showcased the power of Mixed Integer Linear Programming (MILP) in solving cryptanalysis problems that otherwise, required significant effort. Since the inception, research in this area has moved in primarily two directions. One is to model more and more classical cryptanalysis tools as optimization problems to leverage the ease provided by state-of-the-art solvers. The other direction is to improve existing models to make them more efficient and/or accurate. The current work is an attempt to contribute to the latter. In this work, a general model referred to as DEEPAND has been devised to capture the correlation between AND gates in NLFSR-based lightweight block ciphers. DEEPAND builds upon and generalizes the idea of joint propagation of differences through AND gates captured using refined MILP modeling of TinyJAMBU by Sahaet al. in FSE 2020. The proposed model has been applied to TinyJAMBU, KATAN, KTANTAN and can detect correlations that were missed by earlier models. This leads to more accurate differential bounds for both the ciphers. In particular, a 384-round (full-roundas per earlier specification) Type-IV trail is found for TinyJAMBU with 14-active AND gates using the new model, while the refined model reported this figure to be 19. This also reaffirms the decision of the designers to increase the number of rounds from 384 to 640. Moreover, the model succeeds in searching afull roundType-IV trail of TinyJAMBU keyed permutationP1024with probability 2−105(≫ 2−128). This reveals the non-random properties ofP1024thereby showing it to benon-ideal. Hence it cannot be expected to provide the same security levels as robust block ciphers. Further, the provable security of TinyJAMBU AEAD scheme should be carefully revisited. Similarly, for the variants of KATAN, several previously reported trails are improved upon by employing the DEEPAND model. Moreover, in the related-key setting, the DEEPAND model is able to make a better 140-round boomerang distinguisher (for both the data and time complexity) in comparison to the previous boomerang attack by Isobeet al. in ACISP 2013. Furthermore, for enhanced applicability, we employ the DEEPAND model on another multiple AND-based cipher, KTANTAN , in the related-key setting. Our analysis reveals practical differential distinguishers with low data and time complexities for all full-round KTANTAN variants. In summary, DEEPAND seems to capture the underlying correlation better when multiple AND gates are at play and can be adapted to other classes of ciphers as well.

In this paper, inspired by the work of Beyne and Rijmen at CRYPTO 2022, we explore the accurate probability ofd-differential in the fixed-key model. The theoretical foundations of our method are based on a special matrix - quasi-d-differential transition matrix, which is a natural extension of the quasidifferential transition matrix. The role of quasi-d-differential transition matrices in polytopic cryptananlysis is analogous to that of correlation matrices in linear cryptanalysis. Therefore, the fixed-key probability of ad-differential can be exactly expressed as the sum of the correlations of its quasi-d-differential trails. Then we revisit the boomerang attack from a perspective of 3-differential. Different from previous works, the probability of a boomerang distinguisher can be exactly expressed as the sum of the correlations of its quasi-3-differential trails without any assumptions in our work. In order to illustrate our theory, we apply it to the lightweight block cipher GIFT. It is interesting to find the probability of every optimal 3-differential characteristic of an existing 2-round boomerang is zero, which can be seen as an evidence that the security of block ciphers adopting half-round key XOR might be overestimated previously to some extent in differential-like attacks.