A new analysis of small S-boxes based on a new notion of algebraic immunity

Abstract

We analyze five newly emerged cryptographic characteristics and a new notion of algebraic immunity proposed by us of 16 best affine equivalence classes of 4 × 4 S-boxes, as well as the only best affine equivalence class of 3-bit ones. The best class among the 16 classes of 4-bit ones is determined based on our analysis. A strategy to search for good candidates in the best class is also proposed, which can generate many 4 × 4 S-boxes with better cryptographic characteristics than the ones previously proposed. We also obtain many candidates with better cryptographic characteristics than the previously proposed 3-bit S-boxes by a similar method. The implementations of our candidates and the previously proposed ones are also discussed. The discussion shows that our candidates can achieve a good balance between implementation cost and security.