For a given vectorial Boolean function F from F2 to itself it was defined an associated Boolean function γF (a, b) in 2n variables by C. Carlet, P. Charpin, V. Zinoviev in 1998 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 function...

The square map is one of the functions that is used in cryptography. For instance, the square map is used in Rabin encryption scheme, block cipher RC6 and stream cipher Rabbit, in different forms. In this paper we study a special case of the square map, namely the square function modulo a power of two. We obtain probability distribution of the output of this map as a vectorial Boolean function....

We rst show that a Feistel type block cipher is broken if the round function is approximated by a low degree vectorial Boolean function. The proposed attack is a generalization of the higher order diierential attack to a probabilistic one. We next introduce a notion of higher order bent functions in order to prevent our attack. We then show their explicit constructions.

The two most important criteria for vectorial Boolean functions used as S-boxes in block ci-phers are differential uniformity and nonlinearity. Previous work in this field has focused onlyon nonlinearity and a different criterion, autocorrelation. In this paper, we describe the resultsof experiments in using simulated annealing, memetic algorithms, and ant colony optimisation to...

Abstract. The notion of o-polynomial comes from finite projective geometry. In 2011 and later, it has been shown that those objects play an important role in symmetric cryptography and coding theory to design bent Boolean functions, bent vectorial Boolean functions, semi-bent functions and to construct good linear codes. In this note, we characterize o-polynomials by the Walsh transform of the ...

Criteria based on the analysis of the properties of vectorial Boolean functions for selection of substitutions (S-boxes) for symmetric cryptographic primitives are given. We propose an improved gradient descent method for increasing performance of nonlinear vectorial Boolean functions generation with optimal cryptographic properties. Substitutions are generated by proposed method for the most c...

Propagation criteria and resiliency of vectorial Boolean functions are important for cryptographic purpose (see [1], [2], [3], [4], [7], [8], [10], [11] and [16]). Kurosawa , Stoh [8] and Carlet [1] gave a construction of Boolean functions satisfying PC(l) of order k from binary linear or nonlinear codes. In this paper algebraic-geometric codes over GF (2m) are used to modify the Carlet and Kur...

To appear as a chapter of the volume " Boolean Methods and Models " ,

In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from F 2 to Z2t . Explicitly, we describe gbent functions from F 2 to Z2t , which can be seen as a gbent version of Dillon’s PSap class. For the first time, we also introduce the concept of a vectorial gbent function from F 2 to Zmq , and determine the maximal value which m can atta...

In this paper, we study the properties of sum-of-squares indicator vectorial Boolean functions. Firstly, give upper bound $\sum_{u\in \mathbb{F}_2^n,v\in \mathbb{F}_2^m}\mathcal{W}_F^3(u,v)$. Secondly, based on Walsh-Hadamard transform, a secondary construction bent Further, three kinds indicators functions are defined by autocorrelation function and lower bounds derived. Finally, with respect ...