[Show abstract][Hide abstract] ABSTRACT: Bent functions are optimal combinatorial objects. Since their introduction, substantial efforts have been directed toward their study in the last three decades. A complete classification of bent functions is elusive and looks hopeless today, therefore, not only their characterization, but also their generation are challenging problems. This paper is devoted to the construction of bent functions. First, we provide several new effective constructions of bent functions, self-dual bent functions, and antiself-dual bent functions. Second, we provide seven new infinite families of bent functions by explicitly calculating their dual.
IEEE Transactions on Information Theory 07/2014; 60(7):4397-4407. DOI:10.1109/TIT.2014.2320974 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The main topics and interconnections arising in this paper are symmetric cryptography (S-boxes), coding theory (linear codes) and finite projective geometry (hyperovals). The paper describes connections between the two main areas of information theory on the one side and finite geometry on the other side. Bent vectorial functions are maximally nonlinear multi-output Boolean functions. They contribute to an optimal resistance to both linear and differential attacks of those symmetric cryptosystems in which they are involved as substitution boxes (S-boxes). We firstly exhibit new connections between bent vectorial functions and the hyperovals of the projective plane, extending the recent link between bent Boolean functions and the hyperovals. 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. We show that the hyperovals of \(PG_{2}(2^m)\) from finite projective geometry provide new minimal codes (used in particular in secret sharing schemes, to model the access structures) and give rise to multiples of \(2^r\) -ary ( \(r\) being a divisor of \(m\) ) simplex linear codes (whose duals are the perfect \(2^r\) -ary Hamming codes) over an extension field \({\mathbb F}_{2^{r}}\) of \({\mathbb F}_{2^{}}\) . The following diagram gives an indication of the main topics and interconnections arising in this paper.
[Show abstract][Hide abstract] ABSTRACT: In any connected, undirected graph G = (V, E), the distance d(x, y) between two vertices x and y of G is the minimum number of edges in a path linking x to y in G. A sphere in G is a set of the form S
r
(x) = {y ∈ V : d(x, y) = r}, where x is a vertex and r is a nonnegative integer called the radius of the sphere. We first address in this paper the following question: What is the minimum number of spheres with fixed radius r ≥ 0 required to cover all the vertices of a finite, connected, undirected graph G? We then turn our attention to the Hamming Hypercube of dimension n, and we show that the minimum number of spheres with any radii required to cover this graph is either n or n + 1, depending on the parity of n. We also relate the two above problems to other questions in combinatorics, in particular to identifying codes.
[Show abstract][Hide abstract] ABSTRACT: Minimal linear codes are linear codes such that the support of every codeword does not contain the support of another linearly independent codeword. Such codes have applications in cryptography, e.g. to secret sharing. We here study minimal codes, give new bounds and properties and exhibit families of minimal linear codes. We also introduce and study the notion of quasi-minimal linear codes, which is a relaxation of the notion of minimal linear codes, where two non-zero codewords have the same support if and only if they are linearly dependent.
[Show abstract][Hide abstract] ABSTRACT: This paper is devoted to hyperbent functions with multiple trace terms (including binomial functions) via Dillon-like exponents. We show how the approach developed by Mesnager to extend the Charpin–Gong family, which was also used by Wang and coworkers to obtain another similar extension, fits in a much more general setting. 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 and coworkers, but also to characterize the hyperbentness of several new infinite classes of Boolean functions. 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 compare the asymptotic and practical performances of different characterizations, including these in terms of hyperelliptic curves, and actually build hyperbent functions in cases which could not be attained through naive computations of exponential sums.
IEEE Transactions on Information Theory 05/2013; 59(5):3215-3232. DOI:10.1109/TIT.2013.2238580 · 2.33 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 Maiorana-McFarland 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 11/2012; 58(11):6979-6985. DOI:10.1109/TIT.2012.2206557 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper, the relation between binomial Niho bent functions discovered
by Dobbertin et al. and o-polynomials 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
$m\equiv 2 (\bmod 4)$.
[Show abstract][Hide abstract] ABSTRACT: This paper is devoted to hyper-bent functions with multiple trace terms (including binomial functions) via Dillon-like exponents. We show how the approach developed by Mesnager to extend the Charpin-Gong family, which was also used by Wang et al. to obtain another similar extension, fits in a much more general setting. 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 al., but also to characterize the hyper-bentness of new infinite classes of Boolean functions.
[Show abstract][Hide abstract] 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 05/2012; 58(5):3287-3292. DOI:10.1109/TIT.2011.2181330 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This extended abstract is a reduced version of the paper (Carlet and Mesnager 2011). We refer to this paper for the proofs and for complements.
International Symposium on Artificial Intelligence and Mathematics (ISAIM 2012), Fort Lauderdale, Florida, USA, January 9-11, 2012; 01/2012
[Show abstract][Hide abstract] ABSTRACT: In this paper, we study the action of Dickson polynomials on subsets of finite fields of even characteristic related to the trace of the inverse of an element and provide an alternate proof of a not so well-known result. Such properties are then applied to the study of a family of Boolean functions and a characterization of their hyper-bentness in terms of exponential sums recently proposed by Wang et al.Finally, we extend previous works of Lisoněk and Flori and Mesnager to reformulate this characterization in terms of the number of points on hyperelliptic curves and present some numerical results leading to an interesting problem.
[Show abstract][Hide abstract] 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; 57(11-57):7443 - 7458. DOI:10.1109/TIT.2011.2160039 · 2.33 Impact Factor
[Show abstract][Hide abstract] 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 (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. Index Terms—Bent functions, Boolean function, covering ra- dius, cubic sums, Dickson polynomials, hyper-bent functions, Kloosterman sums, maximum nonlinearity, Reed-Muller codes, Walsh-Hadamard transformation.
IEEE Transactions on Information Theory 09/2011; 57(9):5996-6009. DOI:10.1109/TIT.2011.2124439 · 2.33 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Given a set C of q-ary n-tuples and c ∈ C, how many symbols of c suffice to distinguish it from the other elements in C? This is a generalization of an old combinatorial problem, on which we present (asymptotically tight) bounds and variations.
Data Compression, Communications and Processing (CCP), 2011 First International Conference on; 07/2011
[Show abstract][Hide abstract] 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, hyper-bent functions and semi-bent 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–Walsh-Hadamard transform–maximum nonlinearity–bent functions–hyper-bent functions–semi-bent functions
[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 $${s_2={\frac {2^n-1}3}}$$, 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 co-prime with 2
m
+ 1. The corresponding bent functions are also hyper-bent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums.
[Show abstract][Hide abstract] ABSTRACT: Semi-bent functions with even number of variables are a class of important Boolean functions whose Hadamard transform takes three values. Semi-bent functions have been extensively studied due to their applications in cryptography and coding theory. In this paper we are interested in the property of semi-bentness of Boolean functions defined on the Galois field <${\mathbb F}_2^n$ (n even) with multiple trace terms obtained via Niho functions and two Dillon-like functions (the first one has been studied by the author and the second one has been studied very recently by Wang et al. using an approach introduced by the author). We subsequently give a connection between the property of semi-bentness and the number of rational points on some associated hyperelliptic curves. We use the hyperelliptic curve formalism to reduce the computational complexity in order to provide an efficient test of semi-bentness leading to substantial practical gain thanks to the current implementation of point counting over hyperelliptic curves.
[Show abstract][Hide abstract] ABSTRACT: The connection between exponential sums and algebraic varieties has been known for at least six decades. Recently, Lisoněk exploited it to reformulate the Charpin-Gong characterization of a large class of hyper-bent functions in terms of numbers of points on hyperelliptic curves. 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. 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.
Journal of Mathematical Cryptology 01/2011; 2011(1):373. DOI:10.1515/jmc-2012-0013