the upper bounds on differntial characteristics in block cipher SMS4

SMS4 is a 128-bit block cipher with a 128-bit user key and 32 rounds, which is used in the Chinese National Standard for Wireless LAN WAPI. In this paper, all possible differential patterns are divided into several sections by six designed rules. In order to evaluate the security against the differential cryptanalysis of SMS4, we calculate the lower bounds on the number of active S-Boxes for all kinds of sections, based on which the lower bounds on the number of active S-Boxes in all possible differential patterns can be derived. Finally, the upper bounds on differential characteristic probabilities of arbitrary round numbers are given, which can be used to estimate the strength of SMS4 against differential attack and linear attack