A Reversible Transformation, The Number Of Branches

In this paper, results on analysis of branch numbers of Rijndael algorithm and SMS4 algorithm for special invertible linear transformations are generalized to those for arbitrary invertible linear transformations. Firstly, applying matrix theory and recursive method, a criterion theorem is given for any invertible lin-ear transformation on F28×4 to determine linear branch numbers and difference branch numbers. From this theorem, the linear branch number and the dif-ference branch number can be completely determined, while in past works, by linear code theory one can only know whether the two numbers are maximum or not. As a consequence, the results on branch numbers in reference [2] are recovered. Secondly, it is known that for an invertible linear transformation on F28×4 its difference branch number is equal to its linear branch number when it is maximum, say, it is 5. In this paper, the result is further improved as that the difference branch number is equal to its linear branch number, when it is more than 3. Furthermore, for the SMS4 algorithm, based on the intrinsic proper-ties of its matrix of the cyclic shift plus on F2, a simplified form of the above criterion theorem is given. As a corollary, it is proved theoretically that the branch numbers of SMS4 are maximum, while in the past this result was only confirmed through enumeration by computer. In addition, using the knowledge of local rings, a sufficient and necessary condition of whether the cyclic shift plus on F2 is invertible is achieved, precisely, it is proved that the cyclic shift plus on F2 is invertible if and only if the number of nontrivial terms is odd. Finally, the criterion theorem on F28×4 about the branch numbers is extended to the one on F2n×m.