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This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear (n,m) functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions Tr_{m}^{n} (\lambda x^{d}) from \mathbb {F}_{2^{n}} to \mathbb F2^{m} (where m is a divisor of n ) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratović-Ribić et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of (n,m) -functions. No tight upper bound is known when \frac n2

The recent algebraic attacks have received a lot of attention in cryptographic literature. The algebraic immunity of a Boolean function quantifies its resistance to the standard algebraic attacks of the pseudorandom generators using it as a nonlinear filtering or combining function. Very few results have been found concerning its relation with the other cryptographic parameters or with the rth-order nonlinearity. As recalled by Carlet at CRYPTO'06, many papers have illustrated the importance of the rth-order nonlinearity profile (which includes the first-order nonlinearity). The role of this parameter relatively to the currently known attacks has been also shown for block ciphers. Recently, two lower bounds involving the algebraic immunity on the rth-order nonlinearity have been shown by Carlet et al.. None of them improves upon the other one in all situations. In this paper, we prove a new lower bound on the rth-order nonlinearity profile of Boolean functions, given their algebraic immunity, that improves significantly upon one of these lower bounds for all orders and upon the other one for low orders. [ABSTRACT FROM AUTHOR]

An involution over finite fields is a permutation polynomial whose inverse is itself. Owing to this property, involutions over finite fields have been widely used in applications, such as cryptography and coding theory. Following the idea by Wang to characterize the involutory behavior of the generalized cyclotomic mappings, this paper gives a more concise criterion for $x^{r}h(x^{s})\in {\mathbb F} _{q}[x]$ being involutions over the finite field ${\mathbb F}_{q}$ , where $r\geq 1$ and $s\,|\, (q-1)$. By using this criterion, we propose a general method to construct involutions of the form $x^{r}h(x^{s})$ over ${\mathbb F}_{q}$ from given involutions over some subgroups of ${\mathbb F}_{q}^{*}$ by solving congruent and linear equations over finite fields. Then, many classes of explicit involutions of the form $x^{r}h(x^{s})$ over ${\mathbb F}_{q}$ are obtained. [ABSTRACT FROM AUTHOR]

We provide an extensive study of the differential properties of the functions x\mapsto x^2^t-1 over \BBF 2^{n}, for 1 < t< n. We notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials x^2^t+bx^2+(b+1)x where b varies in \BBF 2^{n}. We prove a strong relationship between the differential spectra of x\mapsto x^2^t-1 and x\mapsto x^2^s-1 for s= n-t+1. As a direct consequence, this result enlightens a connection between the differential properties of the cube function and of the inverse function. We also determine the complete differential spectra of x \mapsto x^7 by means of the value of some Kloosterman sums, and of x \mapsto x^2^t-1 for t \in \\lfloor n/2\rfloor , \lceil n /2\rceil +1, n-2\. [ABSTRACT FROM AUTHOR]

A theoretical framework for the use of multiple linear approximations in the linear cryptanalysis of block ciphers is given. The covariance of two mask counts is derived, and it is shown that under appropriate conditions the mask counts in linear cryptanalysis are stochastically independent, whether or not the masks are linearly independent. Some consequences of these observations are also considered. [ABSTRACT FROM AUTHOR]