Sihem Mesnager

Improving the Upper Bounds on the Covering Radii of Binary Reed–Muller Codes

By deriving bounds on character sums of Boolean functions and by using the characterizations, due to Kasami , of those elements of the Reed-Muller codes whose Hamming weights are smaller than twice and a half the minimum distance, we derive an improved upper bound on the covering radius of the Reed-Muller code of order 2, and we deduce improved upper bounds on the covering radii of the Reed-Muller codes of higher orders

Bent and Hyper-Bent Functions in Polynomial Form and Their Link With Some Exponential Sums and Dickson Polynomials

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions, whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials) and give some conjectures that lead to constructions of new hyper-bent functions.

Four decades of research on bent functions

In this survey, we revisit the Rothaus paper and the chapter of Dillon’s thesis dedicated to bent functions, and we describe the main results obtained on these functions during these last 40 years. We also cover more briefly super-classes of Boolean functions, vectorial bent functions and bent functions in odd characteristic.

A New Family of Hyper-Bent Boolean Functions in Polynomial Form

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. These combinatorial objects, with fascinating properties, are rare. The class of bent functions contains a subclass of functions the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer. (Hyper)-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. Few constructions of hyper-bent functions defined over the Galois field

Several New Infinite Families of Bent Functions and Their Duals

{\mathbb F}_{2n}

Hyper-bent Boolean functions with multiple trace terms

(n = 2m ) are proposed in the literature. The known ones are mostly monomial functions. This paper is devoted to the construction of hyper-bent functions. We exhibit an infinite class over

Semibent Functions From Dillon and Niho Exponents, Kloosterman Sums, and Dickson Polynomials

{\mathbb F}_{2n}

Hyperbent Functions via Dillon-Like Exponents

(n = 2m , m odd) having the form

Further Results on Niho Bent Functions

f(x) = Tr_1^{o(s_1)} (a x^{s_1}) + Tr_1^{o(s_2)} (b x^{s_2})

On a conjecture about binary strings distribution

where o (s i ) denotes the cardinality of the cyclotomic class of 2 modulo 2 n *** 1 which contains s i and whose coefficients a and b are, respectively in