The Experts below are selected from a list of 315 Experts worldwide ranked by ideXlab platform
Peter Goddard - One of the best experts on this subject based on the ideXlab platform.
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The Polynomial Form of the Scattering Equations
Journal of High Energy Physics, 2014Co-Authors: Louise Dolan, Peter GoddardAbstract:The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reFormulated in Polynomial Form. The scattering equations for $N$ particles are shown to be equivalent to a Moebius invariant system of $N-3$ equations, $\tilde h_m=0$, $2 \leq m \leq N-2$, in $N$ variables, where $\tilde h_m$ is a homogeneous Polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Moebius invariance appropriately, yields Polynomial equations $h_m=0$, $1 \leq m \leq N-3$, in $N-3$ variables, where $h_m$ has degree $m$. The linearity of the equations in the individual variables facilitates computation, e.g the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the $\tilde h_m$ and $h_m$. The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.
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the Polynomial Form of the scattering equations
Journal of High Energy Physics, 2014Co-Authors: Louise Dolan, Peter GoddardAbstract:The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reFormulated in Polynomial Form. The scattering equations for N particles are shown to be equivalent to a Mobius invariant system of N − 3 equations, $$ \tilde{h} $$ m = 0, 2 ≤ m ≤ N − 2, in N variables, where $$ \tilde{h} $$ m is a homogeneous Polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Mobius invariance appropriately, yields Polynomial equations h m = 0, 1 ≤ m ≤ N − 3, in N − 3 variables, where h m has degree m. The linearity of the equations in the individual variables facilitates computation, e.g. the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the $$ \tilde{h} $$ m and h m . The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.
Sihem Mesnager - One of the best experts on this subject based on the ideXlab platform.
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Further results on semi-bent functions in Polynomial Form
Advances in Mathematics of Communications, 2016Co-Authors: Xiwang Cao, Hao Chen, Sihem MesnagerAbstract:Plateaued functions have been introduced by Zheng and Zhang in 1999 as good candidates for designing cryptographic functions since they possess many desirable cryptographic characteristics. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions ($0$-plateaued functions) and the semi-bent functions ($2$-plateaued functions). Bent functions have been extensively investigated since 1976. Very recently, the study of semi-bent functions has attracted a lot of attention in symmetric cryptography. Many intensive progresses in the design of such functions have been made especially in recent years. The paper is devoted to the construction of semi-bent functions on the finite field $\mathbb{F}_{2^n}$ ($n=2m$) in the line of a recent work of S. Mesnager [IEEE Transactions on InFormation Theory, Vol 57, No 11, 2011]. We extend Mesnager's results and present a new construction of infinite classes of binary semi-bent functions in Polynomial trace. The extension is achieved by inserting mappings $h$ on $\mathbb{F}_{2^n}$ which can be expressed as $h(0) = 0$ and $h(uy) = h_1(u)h_2(y)$ with $u$ ranging over the circle $U$ of unity of $\mathbb{F}_{2^n}$, $y \in \mathbb{F}_{2^m}^{*}$ and $uy \in \mathbb{F}_{2^n}^{*}$, where $h_1$ is a isomorphism on $U$ and $h_2$ is an arbitrary mapping on $\mathbb{F}_{2^m}^{*}$. We then characterize the semi-bentness property of the extended family in terms of classical binary exponential sums and binary Polynomials.
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On Semibent Boolean Functions
IEEE Transactions on Information Theory, 2012Co-Authors: Claude Carlet, Sihem MesnagerAbstract: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.
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bent and hyper bent functions in Polynomial Form and their link with some exponential sums and dickson Polynomials
IEEE Transactions on Information Theory, 2011Co-Authors: Sihem MesnagerAbstract: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.
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a new family of hyper bent boolean functions in Polynomial Form
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding, 2009Co-Authors: Sihem MesnagerAbstract: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 ${\mathbb F}_{2n}$ (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 ${\mathbb F}_{2n}$ (n = 2m , m odd) having the Form $f(x) = Tr_1^{o(s_1)} (a x^{s_1}) + Tr_1^{o(s_2)} (b x^{s_2})$ 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 ${\mathbb F}_{2^{o(s_1)}}$ and ${\mathbb F}_{2^{o(s_2)}}$. We prove that the exponents $s_1={3(2^m-1)}$ and $s_2={\frac {2^n-1}3}$, where $a\in {\mathbb F}_{2n}$ ($a\not=0$) and $b\in{\mathbb F}_4$ provide a construction of hyper-bent functions over ${\mathbb F}_{2n}$ with optimum algebraic degree. We give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums and the cubic sums involving only the coefficient a .
Tor Helleseth - One of the best experts on this subject based on the ideXlab platform.
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new constructions of quadratic bent functions in Polynomial Form
IEEE Transactions on Information Theory, 2014Co-Authors: Xiaohu Tang, Tor HellesethAbstract:New quadratic bent functions in Polynomial Form are constructed in this paper. The constructions give new Boolean bent, generalized Boolean bent and p-ary bent functions. Based on Z 4 -valued quadratic Forms, a simple method provides several new constructions of generalized Boolean bent functions. From these generalized Boolean bent functions a method is presented to transForm them into Boolean bent and semi-bent functions. Moreover, many new p-ary bent functions can also be obtained by applying similar methods.
Louise Dolan - One of the best experts on this subject based on the ideXlab platform.
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The Polynomial Form of the Scattering Equations
Journal of High Energy Physics, 2014Co-Authors: Louise Dolan, Peter GoddardAbstract:The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reFormulated in Polynomial Form. The scattering equations for $N$ particles are shown to be equivalent to a Moebius invariant system of $N-3$ equations, $\tilde h_m=0$, $2 \leq m \leq N-2$, in $N$ variables, where $\tilde h_m$ is a homogeneous Polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Moebius invariance appropriately, yields Polynomial equations $h_m=0$, $1 \leq m \leq N-3$, in $N-3$ variables, where $h_m$ has degree $m$. The linearity of the equations in the individual variables facilitates computation, e.g the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the $\tilde h_m$ and $h_m$. The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.
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the Polynomial Form of the scattering equations
Journal of High Energy Physics, 2014Co-Authors: Louise Dolan, Peter GoddardAbstract:The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reFormulated in Polynomial Form. The scattering equations for N particles are shown to be equivalent to a Mobius invariant system of N − 3 equations, $$ \tilde{h} $$ m = 0, 2 ≤ m ≤ N − 2, in N variables, where $$ \tilde{h} $$ m is a homogeneous Polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Mobius invariance appropriately, yields Polynomial equations h m = 0, 1 ≤ m ≤ N − 3, in N − 3 variables, where h m has degree m. The linearity of the equations in the individual variables facilitates computation, e.g. the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the $$ \tilde{h} $$ m and h m . The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.
Yang Zhang - One of the best experts on this subject based on the ideXlab platform.
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The Polynomial Form of the scattering equations is an H -basis
Physical Review D, 2016Co-Authors: Jorrit Bosma, Mads Søgaard, Yang ZhangAbstract:We prove that the Polynomial Form of the scattering equations is a Macaulay H-basis. We demonstrate that this H-basis facilitates integrand reduction and global residue computations in a way very similar to using a Gr\"obner basis, but circumvents the heavy computation of the latter. As an example, we apply the H-basis to prove the conjecture that the dual basis of the Polynomial scattering equations must contain one constant term.