Publications (32)17.28 Total impact
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ABSTRACT: In this paper, the relation between binomial Niho bent functions discovered by Dobbertin et al. and opolynomials that give rise to the Subiaco and Adelaide classes of hyperovals is found. This allows to expand the class of bent functions that corresponds to Subiaco hyperovals, in the case when $m\equiv 2 (\bmod 4)$.10/2012;  IACR Cryptology ePrint Archive. 01/2012; 2012:20.
 IACR Cryptology ePrint Archive. 01/2012; 2012:33.

Conference Proceeding: On Dillon's class H of Niho bent functions and opolynomials.
International Symposium on Artificial Intelligence and Mathematics (ISAIM 2012), Fort Lauderdale, Florida, USA, January 911, 2012; 01/2012 
Article: On Semibent Boolean Functions
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ABSTRACT: We show that any Boolean function, in even dimension, equal to the sum of a Boolean function $g$ which is constant on each element of a spread and of a Boolean function $h$ whose restrictions to these elements are all linear, is semibent if and only if $g$ and $h$ are both bent. We deduce a large number of infinite classes of semibent functions in explicit bivariate (respectively, univariate) polynomial form.IEEE Transactions on Information Theory 01/2012; 58(5):32873292. · 2.62 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: This paper consists of two main contributions. First, the Niho bent function consisting of 2r exponents (discovered by Leander and Kholosha) is studied. The dual of the function is found and it is shown that this new bent function is not of the Niho type. Second, all known univariate representations of Niho bent functions are analyzed for their relation to the completed MaioranaMcFarland class M. In particular, it is proven that two families do not belong to the completed class M. The latter result gives a positive answer to an open problem whether the class H of bent functions introduced by Dillon in his thesis of 1974 differs from the completed class M.IEEE Transactions on Information Theory 01/2012; 58(11):69796985. · 2.62 Impact Factor 
Article: Semibent Functions From Dillon and Niho Exponents, Kloosterman Sums, and Dickson Polynomials
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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 semibentness 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 semibent functions in even dimension with maximum degree. Moreover, we study the semibentness property of functions in polynomial forms with multiple trace terms and exhibit criteria involving Dickson polynomials.IEEE Transactions on Information Theory 12/2011; · 2.62 Impact Factor 
Chapter: Binary Kloosterman Sums with Value 4
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ABSTRACT: Kloosterman sums have recently become the focus of much research, most notably due to their applications in cryptography and their relations to coding theory. Very recently Mesnager has showed that the value 4 of binary Kloosterman sums gives rise to several infinite classes of bent functions, hyperbent functions and semibent functions in even dimension. In this paper we analyze the different strategies used to find zeros of binary Kloosterman sums to develop and implement an algorithm to find the value 4 of such sums. We then present experimental results showing that the value 4 of binary Kloosterman sums gives rise to bent functions for small dimensions, a case with no mathematical solution so far. KeywordsKloosterman sums–elliptic curves–Boolean functions–WalshHadamard transform–maximum nonlinearity–bent functions–hyperbent functions–semibent functions07/2011: pages 6178; 
Conference Proceeding: On the Link of Some Semibent Functions with Kloosterman Sums.
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ABSTRACT: We extensively investigate the link between the semi bentness property of some Boolean functions in polynomial forms and Kloosterman sums.Coding and Cryptology  Third International Workshop, IWCC 2011, Qingdao, China, May 30June 3, 2011. Proceedings; 01/2011  IACR Cryptology ePrint Archive. 01/2011; 2011:364.
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ABSTRACT: Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were intro duced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (ReedMuller 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 socalled hyperbent functions, whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyperbent 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 hyperbent 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 hyperbent functions. Index Terms—Bent functions, Boolean function, covering ra dius, cubic sums, Dickson polynomials, hyperbent functions, Kloosterman sums, maximum nonlinearity, ReedMuller codes, WalshHadamard transformation.IEEE Transactions on Information Theory 01/2011; 57:59966009. · 2.62 Impact Factor  [show abstract] [hide abstract]
ABSTRACT: Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ (n = 2m) having the form $${f(x) = tr_{o(s_1)} (a x^ {s_1}) + tr_{o(s_2)} (b x^{s_2})}$$ 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 $${F_{2^{o(s_1)}}}$$ and $${F_{2^{o(s_2)}}}$$. Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case. We prove that the exponents s 1 = 2 m − 1 and $${s_2={\frac {2^n1}3}}$$, where $${a\in\mathbb{F}_{2^{n}}}$$ (a ≠ 0) and $${b\in\mathbb{F}_{4}}$$ provide a construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ with optimum algebraic degree. For m odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums. We generalize the result for functions whose exponent s 1 is of the form r(2 m − 1) where r is coprime with 2 m + 1. The corresponding bent functions are also hyperbent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums.Designs Codes and Cryptography 01/2011; 59:265279. · 0.78 Impact Factor 
Article: An efficient characterization of a family of hyperbent functions with multiple trace terms.
IACR Cryptology ePrint Archive. 01/2011; 2011:373. 
Conference Proceeding: Binary Kloosterman Sums with Value 4.
Cryptography and Coding  13th IMA International Conference, IMACC 2011, Oxford, UK, December 1215, 2011. Proceedings; 01/2011  [show abstract] [hide abstract]
ABSTRACT: Computed is the dual of the Niho bent function consisting of 2r exponents that was found by Leander and Kholosha. The algebraic degree of the dual is calculated and it is shown that this new bent function is not of the Niho type. This note is a followup of the recent paper by Carlet and Mesnager.01/2011;  [show abstract] [hide abstract]
ABSTRACT: It is a difficult challenge to find Boolean functions used in stream ciphers achieving all of the necessary criteria and the research of such functions has taken a significant delay with respect to cryptanalyses. Very recently, an infinite class of Boolean functions has been proposed by Tu and Deng having many good cryptographic properties under the assumption that the following combinatorial conjecture about binary strings is true: Conjecture 0.1. Let S t,k be the following set: St,k = {(a,b) Î ( \mathbb Z / (2k1) \mathbb Z )2  a + b = t and w(a) + w(b) < k }. {S_{t,k}}= \left\{{(a,b)} {\in} \left( {{\mathbb Z} / {(2^k1)} {\mathbb Z}} \right)^2  a + b = t ~{\rm and }~ w(a) + w(b) < k \right\}. Then: St,k £ 2k1. {S_{t,k}} \leq 2^{k1}. The main contribution of the present paper is the reformulation of the problem in terms of carries which gives more insight on it than simple counting arguments. Successful applications of our tools include explicit formulas of St,k\left{S_{t,k}}\right for numbers whose binary expansion is made of one block, a proof that the conjecture is asymptotically true and a proof that a family of numbers (whose binary expansion has a high number of 1s and isolated 0s) reaches the bound of the conjecture. We also conjecture that the numbers in that family are the only ones reaching the bound.09/2010: pages 346358;  IACR Cryptology ePrint Archive. 01/2010; 2010:486.

Conference Proceeding: Hyperbent Boolean Functions with Multiple Trace Terms.
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ABSTRACT: Bent functions are maximally nonlinear Boolean functions with an even number of variables. These combinatorial objects, with fascinating properties, are rare. The class of bent functions contains a subclass of functions the socalled hyperbent functions whose properties are still stronger and whose elements are still rarer. In fact, hyperbent functions seem still more difficult to generate at random than bent functions and many problems related to the class of hyperbent functions remain open. (Hyper)bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. In this paper, we contribute to the knowledge of the class of hyperbent functions on finite fields \mathbbF2n\mathbb{F}_{2^n} (where n is even) by studying a subclass \mathfrak Fn\mathfrak {F}_n of the socalled Partial Spreads class PS − (such functions are not yet classified, even in the monomial case). Functions of \mathfrak Fn\mathfrak {F}_n have a general form with multiple trace terms. We describe the hyperbent functions of \mathfrak Fn\mathfrak {F}_n and we show that the bentness of those functions is related to the Dickson polynomials. In particular, the link between the Dillon monomial hyperbent functions of \mathfrak Fn\mathfrak {F}_n and the zeros of some Kloosterman sums has been generalized to a link between hyperbent functions of \mathfrak Fn\mathfrak {F}_n and some exponential sums where Dickson polynomials are involved. Moreover, we provide a possibly new infinite family of hyperbent functions. Our study extends recent works of the author and is a complement of a recent work of Charpin and Gong on this topic.Arithmetic of Finite Fields, Third International Workshop, WAIFI 2010, Istanbul, Turkey, June 2730, 2010. Proceedings; 01/2010  International Journal of Information and Coding Theory 01/2010; 1(2).
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ABSTRACT: One of the classes of bent Boolean functions introduced by John Dillon in his thesis is family H. 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 wellknown Maiorana–McFarland class. We first notice that H can be extended to a slightly larger class that we denote by H. We observe that the bent functions constructed via Niho power functions, for which four examples are known due to Dobbertin et al. and to Leander and Kholosha, are the univariate form of the functions of class H. Their restrictions to the vector spaces ωF2n/2, ω∈F2n⋆, are linear. We also characterize the bent functions whose restrictions to the ωF2n/2ʼs are affine. We answer the open question raised by Dobbertin et al. (2006) in [11] on whether the duals of the Niho bent functions introduced in the paper are affinely equivalent to them, by explicitly calculating the dual of one of these functions. We observe that this Niho function also belongs to the Maiorana–McFarland class, which brings us back to the problem of knowing whether H (or H) is a subclass of the Maiorana–McFarland completed class. We then show that the condition for a function in bivariate form to belong to class H is equivalent to the fact that a polynomial directly related to its definition is an opolynomial (also called oval polynomial, a notion from finite geometry). Thanks to the existence in the literature of 8 classes of nonlinear opolynomials, we deduce a large number of new cases of bent functions in H, which are potentially affinely inequivalent to known bent functions (in particular, to Maiorana–McFarlandʼs functions).IACR Cryptology ePrint Archive. 01/2010; 2010:567.
Publication Stats
88  Citations  
17.28  Total Impact Points  
Top Journals
Institutions

2010–2011

French National Centre for Scientific Research
Lutetia Parisorum, ÎledeFrance, France


2005

Université de Vincennes  Paris 8
SaintDenis, ÎledeFrance, France


2004

Portail des Mathématiques Jussieu / Chevaleret
Lutetia Parisorum, ÎledeFrance, France
