Finding Differential Paths in ARX Ciphers through Nested Monte-Carlo Search

We propose the adaptation of Nested Monte-Carlo Search algorithm for finding differential trails in the class of ARX ciphers. The practical application of the algorithm is demonstrated on round-reduced variants of block ciphers from the SPECK family. More specifically, we report the best differential trails,up to 9 rounds, for SPECK32

Automatic Search for the Best Trails in ARX: Application to Block Cipher Speck

We propose the first adaptation of Matsui's algorithm for finding the best differential and linear trails to the class of ARX ciphers. It is based on a branch-and-bound search strategy, does not use any heuristics and returns optimal results. The practical application of the new algorithm is demonstrated on reduced round variants of block ciphers from the Speck family. More specifically, we report the probabilities of the best differential trails for up to 10, 9, 8, 7, and 7 rounds of Speck32, Speck48, Speck64, Speck96 and Speck128 respectively, together with the exact number of differential trails that have the best probability. The new results are used to compute bounds, under the Markov assumption, on the security of Speck against single-trail differential cryptanalysis. Finally, we propose two new ARX primitives with provable bounds against single-trail differential and linear cryptanalysis -- a long standing open problem in the area of ARX design

MILP-Based Automatic Search Algorithms for Differential and Linear Trails for Speck

In recent years, Mixed Integer Linear Programming (MILP) has been successfully applied in searching for differential characteristics and linear approximations in block ciphers and has produced the significant results for some ciphers such as SIMON (a family of lightweight and hardware-optimized block ciphers designed by NSA) etc. However, in the literature, the MILP-based automatic search algorithm for differential characteristics and linear approximations is still infeasible for block ciphers such as ARX constructions. In this paper, we propose an MILP-based method for automatic search for differential characteristics and linear approximations in ARX ciphers. By researching the properties of differential characteristic and linear approximation of modular addition in ARX ciphers, we present a method to describe the differential characteristic and linear approximation with linear inequalities under the assumptions of independent inputs to the modular addition and independent rounds. We use this representation as an input to the publicly available MILP optimizer Gurobi to search for differential characteristics and linear approximations for ARX ciphers. As an illustration, we apply our method to Speck, a family of lightweight and software-optimized block ciphers designed by NSA, which results in the improved differential characteristics and linear approximations compared with the existing ones. Moreover, we provide the improved differential attacks on Speck48, Speck64, Speck96 and Speck128, which are the best attacks on them in terms of the number of rounds

Differential Cryptanalysis of Round-Reduced SPECK

In this paper, we propose a new algorithm inspired by Nested to find a differential path in ARX ciphers. In order to enhance the decision process of our algorithm and to reduce the search space of our heuristic nested tool, we use the concept of partial difference distribution table (pDDT) along with the algorithm. The algorithm itself is applied on reduced round variants of the SPECK block cipher family. In our previous paper, we applied a naive algorithm with a large search space of values and presented the result only for one block size variant of SPECK. In this new approach, we provide the results within a simpler framework and within a very short period of time for all bigger block size variants of SPECK. More specifically, we report the differential path for up to 8, 9, 11, 10 and 11 rounds of SPECK32, SPECK48, SPECK64, SPECK96 and SPECK128, respectively. To construct a differential characteristics for large number of rounds, we divide long characteristics into short ones, by easily constructing a large characteristic from two short ones. Instead of starting from the first round, we start from the middle and run the experiments forwards as well as in the reverse direction. Using this method, we were able to improve our previous results and report the differential path for up to 9, 10, 12, 13 and 15 rounds of SPECK32, SPECK48, SPECK64, SPECK96 and SPECK128, respectively

Automatic Search for Differential Trails in ARX Ciphers

We propose a tool for automatic search for differential trails in ARX ciphers. By introducing the concept of a partial difference distribution table (pDDT) we extend Matsui's algorithm, originally proposed for DES-like ciphers, to the class of ARX ciphers. To the best of our knowledge this is the first application of Matsui's algorithm to ciphers that do not have S-boxes. The tool is applied to the block ciphers TEA, XTEA, SPECK and RAIDEN. For RAIDEN we find an iterative characteristic on all 32 rounds that can be used to break the full cipher using standard differential cryptanalysis. This is the first cryptanalysis of the cipher in a non-related key setting. Differential trails on 9, 10 and 13 rounds are found for SPECK32, SPECK48 and SPECK64 respectively. The 13 round trail covers half of the total number of rounds. These are the first public results on the security analysis of SPECK. For TEA multiple full (i.e. not truncated) differential trails are reported for the first time, while for XTEA we confirm the previous best known trail reported by Hong et al. We also show closed formulas for computing the exact additive differential probabilities of the left and right shift operations. The source code of the tool is publicly available as part of a larger toolkit for the analysis of ARX at the following address: https://github.com/vesselinux/yaarx

Rotational Cryptanalysis in the Presence of Constants

Rotational cryptanalysis is a statistical method for attacking ARX constructions. It was previously shown that ARX-C, i.e., ARX with the injection of constants can be used to implement any function. In this paper we investigate how rotational cryptanalysis is affected when constants are injected into the state. We introduce the notion of an RX-difference, generalizing the idea of a rotational difference. We show how RX-differences behave around modular addition, and give a formula to calculate their transition probability. We experimentally verify the formula using Speck32/64, and present a 7-round distinguisher based on RX-differences. We then discuss two types of constants: round constants, and constants which are the result of using a fixed key, and provide recommendations to designers for optimal choice of parameters