Classification of Bent Monomials, Constructions of Bent Multinomials and Upper Bounds on the Nonlinearity of Vectorial Functions

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
(λx
^d
) from F
_2
n
to F
_2
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 Muratovic-Ribic 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 n/2 <; m <; n. The covering radius bound is the only known upper bound in this range (the Sidelnikov- Chabaud-Vaudenay bound coincides with it when m = n - 1 and it has no sense when m <; n - 1). Finding better bounds is an open problem since the 1990s. Moreover, no bound has been found during the last 23 years, which improve upon the covering radius bound for a large part of (n, m)-functions. We derive such upper bounds for functions, which are sufficiently unbalanced or which satisfy some conditions. These upper bounds imply some necessary conditions for vectorial functions to have large nonlinearity.