On the nonlinearity of S-boxes and linear codes - Liu, Jian; Mesnager, Sihem; Chen, Lusheng (Cryptography and communications, 2017)
For multi-output Boolean functions (also called S-boxes), various measures of nonlinearity have been widely discussed in the literature but many problems are left open in this topic. The purpose of this paper is to present a new approach to estimating the nonlinearity of S-boxes. A more fine-grained view on the notion of nonlinearity of S-boxes is presented and new connections to some linear codes are established. More precisely, we mainly study the nonlinearity indicator (denoted by 𝒩 v ) for S-boxes from a coding theory point of view. Such a cryptographic parameter 𝒩 v is more related to best affine approximation attacks on stream ciphers. We establish a direct link between 𝒩 v and the minimum distance of the corresponding linear code. We exploit that connection to derive the first general lower bounds on 𝒩 v of non-affine functions from 𝔽 2 n to 𝔽 2 m for m dividing n . Furthermore, we show that 𝒩 v can be determined directly by the weight distribution of the corresponding linear code.