On the estimation of the second largest eigenvalue of Markov ciphers

Abstract

Differential cryptanalysis is an effective tool in modern cryptanalysis. The differential chain of a Markov cipher forms a Markov chain, and the second largest eigenvalue (SLE) of the transition matrix determines the number of iterations such that the Markov cipher can resist differential cryptanalysis. Owing to the huge scale of the transition matrix, it is infeasible to compute the SLE. Thus, an estimation method would be desirable. We find two methods to estimate the SLE by using the elements of the row-stochastic matrix in the literature. Their advantage is parallel computing, without generating the complete matrix. We apply these two methods to the transition matrix of International Data Encryption Algorithm(8) and investigate the accuracy of such estimation. Because the International Data Encryption Algorithm is a primitive Markov cipher, its transition matrix will converge to a uniform distribution. We use the power of the initial transition matrix to estimate the SLE for different number of rounds and compare the results. The errors of the estimation will be acceptable after several rounds when there are less zero elements in the transition matrix and the distribution is more uniform. Moreover, we present a simple relation between the SLE and the number of iterations that the Markov cipher requires against differential cryptanalysis and show the necessary condition of the matrix decomposition method.