This paper evaluates the security of the block cipher E2 against truncated differential cryptanalysis. We show an algorithm to search for effective truncated differentials. The result of the search confirmed that there exist no truncated differentials that lead to possible attacks for E2 with more than 8 rounds. The best attack breaks an 8-round variant of E2 with either IT-Function (the initial transformation) or FT-Function (the final transformation) using 294 chosen plaintexts. We also found the attack which distinguishes a 7-round variant of E2 with IT- and FT-Functions from a random permutation using 291 chosen plaintexts.

We clarify a relation between the XL algorithm and known Grobner basis algorithms. The XL algorithm was proposed to be a more efficient algorithm to solve a system of algebraic equations under a special condition, without calculating a whole Grobner basis. But in our result, it is shown that to solve a system of algebraic equations with a special condition under which the XL algorithm works is equivalent to calculate the reduced Grobner basis of the ideal associated with the system. Moreover we show that the XL algorithm is a Grobner basis algorithm which can be represented as a redundant variant of a known Grobner basis algorithm F4.

This paper describes truncated and impossible differentials of Feistel block ciphers with round functions of 2-layer SPN (Substitution and Permutation Network) transformation modules such as the 128-bit block cipher Camellia, which was proposed by NTT and Mitsubishi Electric Corporation. Our work improves on the best known truncated and impossible differentials, and has found a nontrivial 9-round truncated differential that may lead to a possible attack against a reduced-round version of Camellia without input/output whitening, FL or FL-1 (Camellia-NFL), in the chosen plain text scenario. Previously, only 6-round differentials were known that may suggest a possible attack of Camellia-NFL reduced to 8-rounds. We also show a nontrivial 7-round impossible differential, whereas only a 5-round impossible differential was previously known. We also consider the truncated differential of a reduced-round version of Camellia (Camellia-DS) whose round functions are composed of D-S (Diffusion and Substitution) transformation modules and without input/output whitening, FL or FL-1 (Camellia-DS-NFL), and show a nontrivial 9-round truncated differential, which may lead to a possible attack in the chosen plain text scenario. This truncated differential is effective for general Feistel structures with round functions composed of S-D (Substitution and Diffusion) or D-S transformation.