Nonlinearity Of S-box Research Articles

On the nonlinearity of S-boxes and linear codes

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 ??v$\mathcal {N}_{\mathrm {v}}$) for S-boxes from a coding theory point of view. Such a cryptographic parameter ??v$\mathcal {N}_{\mathrm {v}}$ is more related to best affine approximation attacks on stream ciphers. We establish a direct link between ??v$\mathcal {N}_{\mathrm {v}}$ and the minimum distance of the corresponding linear code. We exploit that connection to derive the first general lower bounds on ??v$\mathcal {N}_{\mathrm {v}}$ of non-affine functions from ??2n$\mathbb {F}_{2^{n}}$ to ??2m$\mathbb {F}_{2^{m}}$ for m dividing n. Furthermore, we show that ??v$\mathcal {N}_{\mathrm {v}}$ can be determined directly by the weight distribution of the corresponding linear code.

On Nonlinearity of S‐Boxes and Their Related Binary Codes

The nonlinearity of S-boxes and their related supercodes of the first order Reed-Muller code are discussed. Based on the properties of multi-output bent functions and almost bent functions, we determine the maximum size of linear supercodes of the first order Reed-Muller code which have optimal or suboptimal minimum distance, and we also give the weight distributions of these supercodes which achieve the best possible size. Furthermore, an upper bound on the minimum distance of a class of binary linear codes is presented, which yields a new upper bound on the nonlinearity of S-boxes. The new bound on nonlinearity improves a bound given by Carlet et al. in 2007.

On the approximation of S‐boxes via Maiorana–McFarland functions

Substitution boxes (S-boxes) are the key components of conventional cryptographic systems. To quantify the confusion property of S-boxes, different non-linearity criteria are proposed such as usual non-linearity (NF ), unrestricted non-linearity (UN F ), generalised non-linearity (GN F ), higher order non-linearity (HN F ) and so on. Although these different criteria come from the idea of linear (or non-linear) approximation of S-boxes, the algebraic structures of Boolean functions that are used to approximate to S-boxes have not been considered yet. In this study, the concept of the extended non-linearity of S-boxes (denoted by EN F ) is introduced by measuring the distance of a given function to a subset of Maiorana–McFarland functions. This approximation appears to be appealing because of a particular structure of this class of functions, namely their representation as a concatenation of affine functions. The complexity of computing the rth order extended non-linearity for S-boxes over GF(2) n is less than O(( n r )2 n−r ), (r > 1). Moreover, a theoretical upper bound for the rth order extended non-linearity is proved, which is much lower than previous generalised non-linearity which might give a rise to more efficient attacks that combine a generalised correlation approach with guess and determine techniques. Furthermore, the relationship between the r-order extended non-linearity and the generalised non-linearity is derived.

Relating Differential Distribution Tables to Other Propertiesof of Substitution Boxes

Due to the success of differential and linear attacks on a large number of encryption algorithms, it is important to investigate relationships among various cryptographic, including differential and linear, characteristics of an S-box (substitution box). After discussing a precise relationship among three tables, namely the difference, auto-correlation and correlation immunity distribution tables, of an S-box, we develop a number of results on various properties of S-boxes. More specifically, we show (1) close connections among three indicators of S-boxes, (2) a tight lower bound on the sum of elements in the leftmost column of its differential distribution table, (3) a non-trivial and tight lower bound on the differential uniformity of an S-box, and (4) two upper bounds on the nonlinearity of S-boxes (one for a general, not necessarily regular, S-box and the other for a regular S-box).