Efficient high nonlinearity S-box generating algorithm based on third-order nonlinear digital filter

In this paper, we first design a third-order nonlinear digital filter (3rd-NDF) with twos complement arithmetic, its trajectories under discrete sine inputs are theoretically analyzed. Analysis results show the trajectories can be partitioned into three categories according to the periodicity of symbolic sequences and chaotic behavior can exhibit when the symbolic sequences are aperiodic. According to its Lyapunov exponents and statistical properties, we find not only the chaotic behaviors of the filter are better than many excellent chaotic systems, but also it has good pseudo-randomness. Then, an S-box generation algorithm based on the filter is presented, in which two Lemmas are presented and proven to construct bijective S-boxes with high nonlinearity. To best of our knowledge, similar methods have never been used in the existing studies. In addition, the designed S-boxes show good cryptographic performances in terms of strict avalanche criteria, differential uniformity, bits independence criterion and linear approximation probability. Finally, we present a novel block cipher algorithm based on the chaotic S-boxes. Analysis results show it is competitive with some of the most advanced algorithms.