Maximum correlation analysis of nonlinear s-boxes in stream ciphers

Abstract

This paper investigates the design of S-boxes used for combining linear feedback shift register (LFSR) sequences in combination generators. Such combination generators have higher throughput than those using Boolean functions as the combining functions. However, S-boxes tend to leak more information about the LFSR sequences than Boolean functions. To study the information leakage, the notion of maximum correlation is introduced, which is based on the correlation between linear functions of the input and all the Boolean functions (linear and nonlinear) of the output of an S-box. Using Walsh transform, a spectral characterization of the maximum correlation coefficients, together with their upper and lower bounds, are established. For the perfect nonlinear S-boxes designed for block ciphers, an upper bound on the maximum correlation coefficients is presented.