On Cryptographically Significant Mappings over GF(2ⁿ)

Abstract

In this paper we investigate the algebraic properties of important cryptographic primitives called substitution boxes (S-boxes). An S-box is a mapping that takes n binary inputs whose image is a binary m-tuple; therefore it is represented as $F:\text{GF}(2)^n \rightarrow \text{GF}(2)^m$. One of the most important cryptographic applications is the case n = m, thus the S-box may be viewed as a function over $\text{GF}(2^n)$. We show that certain classes of functions over $\text{GF}(2^n)$ do not possess a cryptographic property known as APN (Almost Perfect Nonlinear) permutations. On the other hand, when n is odd, an infinite class of APN permutations may be derived in a recursive manner, that is starting with a specific APN permutation on $\text{GF}(2^k)$, k odd, APN permutations are derived over $\text{GF}(2^{k+2i})$ for any i ≥ 1. Some theoretical results related to permutation polynomials and algebraic properties of the functions in the ring $\text{GF}(q)[x,y]$ are also presented. For sparse polynomials over the field $\text{GF}(2^n)$, an efficient algorithm for finding low degree I/O equations is proposed.