Improving Matsui\u27s Search Algorithm for the Best Differential/Linear Trails and its Applications for DES, DESL and GIFT

Automatic search methods have been widely used for cryptanalysis of block ciphers, especially for the most classic cryptanalysis methods -- differential and linear cryptanalysis. However, the automatic search methods, no matter based on MILP, SMT/SAT or CP techniques, can be inefficient when the search space is too large. In this paper, we improve Matsui\u27s branch-and-bound search algorithm which is known as the first generic algorithm for finding the best differential and linear trails by proposing three new methods. The three methods, named Reconstructing DDT and LAT According to Weight, Executing Linear Layer Operations in Minimal Cost and Merging Two 4-bit S-boxes into One 8-bit S-box respectively, can efficiently speed up the search process by reducing the search space as much as possible and reducing the cost of executing linear layer operations. We apply our improved algorithm to DESL and GIFT, which are still the hard instances for the automatic search methods. As a result, we find the best differential trails for DESL (up to 14 rounds) and GIFT-128 (up to 19 rounds). The best linear trails for DESL (up to 16 rounds), GIFT-128 (up to 10 rounds) and GIFT-64 (up to 15 rounds) are also found. To the best of our knowledge, these security bounds for DESL and GIFT under single-key scenario are given for the first time. Meanwhile, it is the longest exploitable (differential or linear) trails for DESL and GIFT. Furthermore, benefiting from the efficiency of the improved algorithm, we do experiments to demonstrate that the clustering effect of differential trails for 13-round DES and DESL are both weak

A new method for Searching Optimal Differential and Linear Trails in ARX Ciphers

In this paper, we propose an automatic tool to search for optimal differential and linear trails in ARX ciphers. It\u27s shown that a modulo addition can be divided into sequential small modulo additions with carry bit, which turns an ARX cipher into an S-box-like cipher. From this insight, we introduce the concepts of carry-bit-dependent difference distribution table (CDDT) and carry-bit-dependent linear approximation table (CLAT). Based on them, we give efficient methods to trace all possible output differences and linear masks of a big modulo addition, with returning their differential probabilities and linear correlations simultaneously. Then an adapted Matsui\u27s algorithm is introduced, which can find the optimal differential and linear trails in ARX ciphers. Besides, the superiority of our tool\u27s potency is also confirmed by experimental results for round-reduced versions of HIGHT and SPECK. More specifically, we find the optimal differential trails for up to 10 rounds of HIGHT, reported for the first time. We also find the optimal differential trails for 10, 12, 16, 8 and 8 rounds of SPECK32/48/64/96/128, and report the provably optimal differential trails for SPECK48 and SPECK64 for the first time. The optimal linear trails for up to 9 rounds of HIGHT are reported for the first time, and the optimal linear trails for 22, 13, 15, 9 and 9 rounds of SPECK32/48/64/96/128 are also found respectively. These results evaluate the security of HIGHT and SPECK against differential and linear cryptanalysis. Also, our tool is useful to estimate the security in the design of ARX ciphers

A New Classification of 4-bit Optimal S-boxes and its Application to PRESENT, RECTANGLE and SPONGENT

In this paper, we present a new classification of 4-bit optimal S-boxes. All optimal 4-bit S-boxes can be classified into 183 different categories, among which we specify 3 platinum categories. Under the design criteria of the PRESENT (or SPONGENT) S-box, there are 8064 different S-boxes up to adding constants before and after an S-box. The 8064 S-boxes belong to 3 different categories, we show that the S-box should be chosen from one out of the 3 categories or other categories for better resistance against linear cryptanalysis. Furthermore, we study in detail how the S-boxes in the 3 platinum categories influence the security of PRESENT, RECTANGLE and SPONGENT88 against differential and linear cryptanalysis. Our results show that the S-box selection has a great influence on the security of the schemes. For block ciphers or hash functions with 4-bit S-boxes as confusion layers and bit permutations as diffusion layers, designers can extend the range of S-box selection to the 3 platinum categories and select their S-box very carefully. For PRESENT, RECTANGLE and SPONGENT88 respectively, we get a set of potentially best/better S-box candidates from the 3 platinum categories. These potentially best/better S-boxes can be further investigated to see if they can be used to improve the security-performance tradeoff of the 3 cryptographic algorithms

Improved (Related-key) Differential Cryptanalysis on GIFT

In this paper, we reevaluate the security of GIFT against differential cryptanalysis under both single-key scenario and related-key scenario. Firstly, we apply Matsui\u27s algorithm to search related-key differential trails of GIFT. We add three constraints to limit the search space and search the optimal related-key differential trails on the limited search space. We obtain related-key differential trails of GIFT-64/128 for up to 15/14 rounds, which are the best results on related-key differential trails of GIFT so far. Secondly, we propose an automatic algorithm to increase the probability of the related-key boomerang distinguisher of GIFT by searching the clustering of the related-key differential trails utilized in the boomerang distinguisher. We find a 20-round related-key boomerang distinguisher of GIFT-64 with probability 2^-58.557. The 25-round related-key rectangle attack on GIFT-64 is constructed based on it. This is the longest attack on GIFT-64. We also find a 19-round related-key boomerang distinguisher of GIFT-128 with probability 2^-109.626. We propose a 23-round related-key rectangle attack on GIFT-128 utilizing the 19-round distinguisher, which is the longest related-key attack on GIFT-128. The 24-round related-key rectangle attack on GIFT-64 and 22-round related-key boomerang attack on GIFT-128 are also presented. Thirdly, we search the clustering of the single-key differential trails. We increase the probability of a 20-round single-key differential distinguisher of GIFT-128 from 2^-121.415 to 2^-120.245. The time complexity of the 26-round differential attack on GIFT-128 is improved from 2^124:415 to 2^123:245

Structural Evaluation of AES and Chosen-Key Distinguisher of 9-round AES-128

While the symmetric-key cryptography community has now a good experience on how to build a secure and efficient fixed permutation, it remains an open problem how to design a key-schedule for block ciphers, as shown by the numerous candidates broken in the related-key model or in a hash function setting. Provable security against differential and linear cryptanalysis in the related-key scenario is an important step towards a better understanding of its construction. Using a structural analysis, we show that the full AES-128 cannot be proven secure unless the exact coefficients of the MDS matrix and the S-Box differential properties are taken into account since its structure is vulnerable to a related-key differential attack. We then exhibit a chosen-key distinguisher for AES-128 reduced to 9 rounds, which solves an open problem of the symmetric community. We obtain these results by revisiting algorithmic theory and graph-based ideas to compute all the best differential characteristics in SPN ciphers, with a special focus on AES-like ciphers subject to related-keys. We use a variant of Dijkstra\u27s algorithm to efficiently find the most efficient related-key attacks on SPN ciphers with an algorithm linear in the number of rounds

Towards Finding the Best Characteristics of Some Bit-oriented Block Ciphers and Automatic Enumeration of (Related-key) Differential and Linear Characteristics with Predefined Properties

In this paper, we investigate the Mixed-integer Linear Programming (MILP) modelling of the differential and linear behavior of a wide range of block ciphers. We point out that the differential behavior of an arbitrary S-box can be exactly described by a small system of linear inequalities. ~~~~~Based on this observation and MILP technique, we propose an automatic method for finding high probability (related-key) differential or linear characteristics of block ciphers. Compared with Sun {\it et al.}\u27s {\it heuristic} method presented in Asiacrypt 2014, the new method is {\it exact} for most ciphers in the sense that every feasible 0-1 solution of the MILP model generated by the new method corresponds to a valid characteristic, and therefore there is no need to repeatedly add valid cutting-off inequalities into the MILP model as is done in Sun {\it et al.}\u27s method; the new method is more powerful which allows us to get the {\it exact lower bounds} of the number of differentially or linearly active S-boxes; and the new method is more efficient which allows to obtain characteristic with higher probability or covering more rounds of a cipher (sometimes with less computational effort). ~~~~~Further, by encoding the probability information of the differentials of an S-boxes into its differential patterns, we present a novel MILP modelling technique which can be used to search for the characteristics with the maximal probability, rather than the characteristics with the smallest number of active S-boxes. With this technique, we are able to get tighter security bounds and find better characteristics. ~~~~~Moreover, by employing a type of specially constructed linear inequalities which can remove {\it exactly one} feasible 0-1 solution from the feasible region of an MILP problem, we propose a method for automatic enumeration of {\it all} (related-key) differential or linear characteristics with some predefined properties, {\it e.g.}, characteristics with given input or/and output difference/mask, or with a limited number of active S-boxes. Such a method is very useful in the automatic (related-key) differential analysis, truncated (related-key) differential analysis, linear hull analysis, and the automatic construction of (related-key) boomerang/rectangle distinguishers. ~~~~~The methods presented in this paper are very simple and straightforward, based on which we implement a Python framework for automatic cryptanalysis, and extensive experiments are performed using this framework. To demonstrate the usefulness of these methods, we apply them to SIMON, PRESENT, Serpent, LBlock, DESL, and we obtain some improved cryptanalytic results