Vol 59 5 , pages , Abstract : This paper is devoted to hyper-bent functions with multiple trace terms including binomial functions via Dillon-like exponents. To this end, we first explain how the original restriction for Charpin--Gong criterion can be weakened before generalizing the Mesnager approach to arbitrary Dillon-like exponents. Afterward, we tackle the problem of devising infinite families of extension degrees for which a given exponent is valid and apply these results not only to reprove straightforwardly the results of Mesnager and Wang et. We go into full details only for a few of them, but provide an algorithm and the corresponding software to apply this approach to an infinity of other new families.

Finally, we propose a reformulation of such characterizations in terms of hyperelliptic curves and use it to actually build hyper-bent functions in cases which could not be attained through naive computations of exponential sums. Further results on Niho bent functions , L.

## AMS :: Mathematics Calendar

Budaghyan, C. Carlet, T. Helleseth, A. Kholosha and S. Vol 58, No 11, pages , The algebraic degree of the dual is calculated and it is shown that this new bent function is not of the Niho type. Finally, three infinite classes of Niho bent functions are analyzed for their relation to the completed Maiorana-McFarland class. This is done using the criterion based on second-order derivatives of a function.

On Semi-bent Boolean Functions , C. Vol 58, No 5, pages: , We deduce a large number of infinite classes of semi-bent functions in explicit bivariate resp. Vol 57, No 11, pages , Abstract : Kloosterman sums have recently become the focus of much research, most notably due to their applications in cryptography and coding theory. In this paper, we extensively investigate the link between the semi-bentness property of functions in univariate forms obtained via Dillon and Niho functions and Kloosterman sums.

In particular, we show that zeros and the value four of binary Kloosterman sums give rise to semi-bent functions in even dimension with maximum degree. Moreover, we study the semi-bentness property of functions in polynomial forms with multiple trace terms and exhibit criteria involving Dickson polynomials. On Dillon's class H of bent functions, Niho bent functions and o-polynomials , C. While this class corresponds to a nice original construction of bent functions in bivariate form, Dillon could exhibit in it only functions which already belonged to the well-known Maiorana-McFarland class.

We observe that the bent functions constructed via Niho power functions, for which four examples are known due to Dobbertin et al. We answer the open question raised by Dobbertin et al. Bent and Hyper-bent functions in polynomial form and their link with some exponential sums and Dickson Polynomials , S.

## Commutative Algebra

Vol 57, No 9, pages , They were introduced by Rothaus in 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 , 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.

A new class of bent and hyper-bent Boolean functions in polynomial forms , S. Mesnager, Journal Designs, Codes and Cryptography. Volume 59, Numbers , pages Abstract : Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case.

The corresponding bent functions are also hyper-bent. On the construction of bent vectorial functions , C. Vol 1, No. Abstract : This paper is devoted to the constructions of bent vectorial functions, that is, maximally nonlinear multi-output Boolean functions. Such functions contribute to an optimal resistance to both linear and differential attacks of those cryptosystems in which they are involved as substitution boxes S-boxes. We survey, study more in details and generalize the known primary and secondary constructions of bent functions, and we introduce new ones.

Abstract : 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.

### Papers and Preprints

As recalled by Carlet at CRYPTO'06, many papers have illustrated the importance of the r th-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. 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. On the number of resilient Boolean functions , S. Abstract : Boolean functions are very important primitives of symmetric cryptosystems. To increase the security of such cryptopsystems, these Boolean functions have to fit several security criteria.

This class of Boolean function has been widely studied by cryptographers. In this paper, we propose a new approach to this question. We reword this question in that to count integer solutions of a system of linear inequalities. Abstract : 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 Test of epimorphism for finitely generated morphisms between affine algebras over Computational rings , S.

Mesnager, Journal of Algebra and Applications, Vol 4 4 , pages Construction of the integral closure of an affine domain in a finite field extension of its quotient field , S. Abstract : The construction of the normalization of an affine domain over a field is a classical problem solved since sixteen's by Stolzenberg and Seidenberg thanks to classical algebraic methods and more recently by Vasconcelos and de Jong thanks to homological methods. As application of our construction, we explain how to obtain an effective decomposition of a quasi-finite and dominant morphism from a normal affine irreducible variety to an affine irreducible variety as a product of an open immersion and a finite morphism, conformly to the classical Grothendieck's version of Zariski's main theorem.

On resultant criteria and formulas for the inversion of a polynomial map , S. Mesnager, Communications in Algebra 29 8 , pages Proceedings of international conferences: in reverse chronological order Strongly regular graphs from weakly regular plateaued functions. Mesnager and A.

Sinak, Proceedings of Ninth International Workshop on Signal Design and its Applications in Communications IWSDA , China Abstract : This paper presents the first construction of strongly regular graphs and association schemes from weakly regular plateaued functions over finite fields of odd characteristic. Indeed, we generalize the construction method of strongly regular graphs from weakly regular bent functions given by Chee et al.

In this framework, we construct strongly regular graphs with three types of parameters from weakly regular plateaued functions with some homogeneous conditions.

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## Commutative Ring Theory and Applications

We also construct a family of association schemes of class p from weakly regular p-ary plateaued functions. Li, S. Very recently, Mesnager and Qu [9] provided a systematic study of 2-to- 1 mappings over finite fields. In addition, another research direction is to consider 2-to- 1 polynomials with few terms.

Some results about 2-to-1 monomials and binomials can be found in [9]. Motivated by their work, in this present paper, we continue to study 2-to-1 mappings, particularly, over finite fields with characteristic 2. Constructions of optimal locally recoverable codes via Dickson polynomials. Mesnager, and D. In , Liu et al. The well-known Dickson polynomials form an important class of polynomials which have been extensively investigated in recent years under different contexts.

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