Automated Differential-Linear Cryptanalysis for AND-RX Ciphers

Abstract

Differential and linear cryptanalysis are two important methods to evaluate the security of block ciphers. Building on these two methods, differential-linear (DL) cryptanalysis was introduced by Langford and Hellman in 1994. This cryptanalytic method has been not only extensively researched but also proven to be effective. In this paper, a security evaluation framework for AND-RX ciphers against DL cryptanalysis is proposed, which is denoted as K6. In addition to modeling the structure of all the possible differential trails and linear trails at the bit level, we introduce a method to calculate this structure round by round. Based on this approach, an automatic algorithm is proposed to construct the DL distinguisher. Unlike previous methods, K6 uses a truncated differential and a linear hull instead of a differential characteristic and a linear approximation, which brings the bias of the DL distinguisher close to the experimental value. To validate the effectiveness of the framework, K6 is applied to Simon and Simeck, which are two typical AND-RX ciphers. With the automatic algorithm, we discover an 11-round DL distinguisher of Simon32 with bias 2−14.89 and a 12-round DL distinguisher of Simeck32 with bias 2−14.89. Moreover, the 14-round DL distinguisher of Simon48 with bias 2−22.30 is longer than the longest DL distinguisher currently known. In addition, the framework K6 shows advantages when analyzing ciphers with large block sizes. As far as we know, for Simon64/96/128 and Simeck48/64, the first DL distinguishers are obtained with our framework. The DL distinguishers are 16, 23, 32, 17, and 22 rounds of Simon64/96/128 and Simeck48/64 with bias 2−24.31, 2−47.57, 2−60.75, 2−22.54, and 2−31.41, respectively. To prove the correctness of distinguishers, experiments on Simon32 and Simeck32 have been performed. The experimental bias are 2−13.76 and 2−14.82, respectively. Comparisons of the theoretical and experimental results show good agreement.