On the sum-of-squares of differential distribution table for (n,n)-functions

Abstract

S-box is one of the core components of symmetric cryptographic algorithms, but differential distribution table (DDT) is an important tool to research some properties of S-boxes to resist differential attacks. In this paper, we give a relationship between the sum-of-squares of DDT and the sum-of-squares indicator of (n,m)-functions based on the autocorrelation coefficients. We also get the upper and lower bounds on the sum-of-squares of DDT of balanced (n,m)-functions, and prove that the sum-of-squares of DDT of (n,m)-functions is affine invariant under affine affine equivalent. Furthermore, we obtain a relationship between the sum-of- squares of DDT and the signal-to-noise ratio of (n,m)-functions. In addition, we calculate the distributions of the sum-of-squares of DDT for all 3-bit S-boxes, the 4-bit optimal S-boxes and all 302 balanced S-boxes (up to affine equivalence), data experiments verify our results.