Vectorial Boolean functions play an important role in cryptography, sequences and coding theory. Both affine equivalence EA-equivalence are well known relations between vectorial functions. In this paper, we give exact formula for the number of classes, asymptotic classes

We give a geometric characterization of vectorial Boolean functions with differential uniformity ≤ 4. This enables us to give a necessary condition on the degree of the base field for a function of degree 2r − 1 to be differentially 4-uniform.

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding et al. recently. Fir...

In [13] for a given vectorial Boolean function F from F2 to itself it was defined an associated Boolean function γF (a, b) in 2n variables that takes value 1 iff a 6= 0 and equation F (x) + F (x + a) = b has solutions. In this paper we introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. It is an interesting open proble...

Boolean functions and vectorial are the most important nonlinear components of stream ciphers. They should satisfy several criteria such as high nonlinearity, proper resiliency so on to guarantee security whole system. However, there some constraints among criteria, how achieve a trade-off between them is an issue. In this paper, possessing simple algebraic normal form with special Walsh spectr...

In this paper we study the relationship between vectorial (Boolean) functions and cyclic codes in the context of algebraic attacks. We first derive a direct link between the annihilators of a vectorial function (in univariate form) and certain 2n-ary cyclic codes (which we prove that they are LCD codes) extending results due to Rønjom and Helleseth. The knowledge of the minimum distance of thos...

Abstract Various systematic modifications of vectorial Boolean functions have been used for finding new previously unknown classes S-boxes with good or even optimal differential uniformity and nonlinearity. In this paper, a general modification method is given that preserves the bijectivity property function in case inverse admits linear structure. A known construction such based on bijective G...

This paper proposes a reversible, balanced and nonlinear vectorial Boolean function called ‘Rain’. Traditional integer addition modulo 2 has several features like reversibility, balancedness and nonlinearity. However, the bias for the best linear approximation of the output bits and their linear combinations is quite high. This leads to several attacks on stream cipher like NLS, which employs a...