Cryptanalysis of Rijndael S-box and improvement

By the discovered correlation between linear functions over GF(qⁿ) and matrices over GF(q), a new scheme is presented to resolve the algebraic expression of Rijndael S-box in this paper. This new scheme has the advantage of predetermining in the case of a given random basis over GF(qⁿ). The reason why only nine terms are involved in the algebraic expression of Rijndael S-box is presented, which corrects the available inaccurate illustration. An improved AES S-box is presented to improve the complexity of AES S-box algebraic expression with terms increasing from 9 to 255 and algebraic degree invariable. The improved AES S-box also has good properties of Boolean functions in SAC and balance, and is capable of attacking against differential cryptanalysis with high reliable security. We finally summarize all the available methods to determine the algebraic expression of Rijndael S-box.