Such a link provides several new classes of optimal vectorial bent functions. Secondly, we exhibit surprisingly a connection between the hyperovals of the projective plane in even characteristic and q-ary simplex codes. To this end, we present a general construction of classes of linear codes from o-polynomials and study their weight distribution proving that all of them are constant weight codes. Bent functions from spreads , S. Abstract : Bent functions are optimal combinatorics objects. Since the introduction of these functions, substantial efforts have been directed towards their study in the last three decades.
The study of such bent functions motivates the clarification of connections between various subclasses of the class of partial bent functions and relations to the class of hyper-bent functions. We investigate their logic relations and state results giving more insight.
We also draw a Venn diagram which explains the relations between these classes. Several new infinite families of bent functions and their duals , S. Abstract : Bent functions are optimal combinatorial objects. A complete classification of bent functions is elusive and looks hopeless today, therefore, not only their characterization, but also their generation are challenging problems.
The paper is devoted to the construction of bent functions. Firstly we provide several new effective constructions of bent functions, self-dual bent functions and anti-self-dual bent functions. Secondly, we provide seven new infinite families of bent functions by explicitly calculating their dual functions.
Sphere coverings and Identifying Codes , D. Auger, G.
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Cohen and S. We also relate the two above problems to other questions in combinatorics, in particular to identifying codes.
On constructions of semi-bent functions from bent functions , G. Abstract : Plateaued functions are significant in cryptography as they possess various desirable cryptographic properties. Two important classes of plateaued functions are those of bent functions and semi-bent functions, due to their combinatorial and algebraic properties.
Constructions of bent functions have been extensively investigated. However only few constructions of semi-bent functions have been proposed in the literature. In general, finding new constructions of bent and semi-bent functions is not a simple task. The paper is devoted to the construction of semi-bent functions with even number of variables. We show that bent functions give rise to primary and secondary-like constructions of semi-bent functions.
An efficient characterization of a family of hyper-bent functions with multiple trace terms , J. Flori and S. Mesnager, Journal of Mathematical Cryptology. Vol 7 1 , pages , Abstract : The connection between exponential sums and algebraic varieties has been known for at least six decades. As a consequence, he obtained a polynomial time and space algorithm for certain subclasses of functions in the Charpin--Gong family. In this paper, we settle a more general framework, together with detailed proofs, for such an approach and show that it applies naturally to a distinct family of functions proposed by Mesnager.
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Doing so, a polynomial time and space test for the hyper-bentness of functions in this family is obtained as well. Nonetheless, a straightforward application of such results does not provide a satisfactory criterion for explicit generation of functions in the Mesnager family. To address this issue, we show how to obtain a more efficient test leading to a substantial practical gain. We finally elaborate on an open problem about hyperelliptic curves related to a family of Boolean functions studied by Charpin and Gong.
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Hyper-bent functions via Dillon-like exponents , S. Mesnager and J. 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. 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.
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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. 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.
Lectures notes ...
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.