Automated Differential-Linear Cryptanalysis for AND-RX Ciphers

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 $$. In addition to modeling the structure of all the possible differential trails and linear trails at the bit level, we introduce a method to calculate this structure round by round. Based on this approach, an automatic algorithm is proposed to construct the DL distinguisher. Unlike previous methods, $$ 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, $$ is applied to Simon and Simeck, which are two typical AND-RX ciphers. With the automatic algorithm, we discover an 11-round DL distinguisher of Simon32 with bias 2^−14.89 and a 12-round DL distinguisher of Simeck32 with bias 2^−14.89. Moreover, the 14-round DL distinguisher of Simon48 with bias 2^−22.30 is longer than the longest DL distinguisher currently known. In addition, the framework $$ 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.