On Some Mathematical Problems In Advanced Encryption Standard

Mathematical theory is the theoretical foundation of cryptography. Further research on mathematics is necessary to ensure the security of cryptographic algorithms. It is introduced in this paper that the history of cryptography and the important role of mathematical theory in cryptography. Then Advanced Encryption Standard is fully detailed. It is researched that algebraic properties of AES and irreducible polynomials over finite fields. The main results are as follows:The polynomial representation of S-boxes being a Boolean function and the branch of mixcolumn transform are researched. According to Small items of Boolean functions, a method to calculate polynomial representation of Boolean functions is presented, using which the algebraic degree of Boolean functions can be figured out quickly. Next, algebraic properties of mixcolumn transform are analyzed. Several definitions of special mixcolumn transform are described, which branches are analyzed. A Method to construct mixcolumn transform with maximum branch under certain conditions is given.The correlation between the order of a polynomial over finite fields and the order of the multiplicative group of the extension field is analyzed. And sufficient and necessary conditions on judging whether or not a polynomial over finite fields is irreducible are presented. Then an efficient new method to determine irreducible polynomials over finite fields is proposed.