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Beside more or less classical mathematical functions, numerical computations for GLMT require accurate enough computations of BSCs \(g_{n}^{m}\) or g n describing the incident beam. These BSCs may be in fact, at least in some cases, experimentally evaluated on an actual beam in a laboratory [328], [329], [330], [331], [332], [333] but, in this book, we mostly discuss the evaluation of BSCs from a priori, or better said assumed, known mathematical descriptions of the incident beam. Then, because a huge amount of BSCs must be evaluated before assembling GLMT-series, much effort has been devoted to the development of efficient schemes and algorithms. This chapter is devoted to the most efficient technique available up to now, namely the localized approximation relying on a localized interpretation of the BSCs, and leading to the construction of localized beam models exactly satisfying Maxwell’s equations.

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 so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer. In fact, hyper-bent functions seem still more difficult to generate at random than bent functions and many problems related to the class of hyper-bent 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 hyper-bent functions on finite fields F2n (where n is even) by studying a subclass Fn of the so-called Partial Spreads class PS- (such functions are not yet classified, even in the monomial case). Functions of Fn have a general form with multiple trace terms. We describe the hyper-bent functions of Fn and we show that the bentness of those functions is related to the Dickson polynomials. In particular, the link between the Dillon monomial hyper-bent functions of Fn and the zeros of some Kloosterman sums has been generalized to a link between hyper-bent functions of Fn and some exponential sums where Dickson polynomials are involved. Moreover, we provide a possibly new infinite family of hyper-bent functions. Our study extends recent works of the author and is a complement of a recent work of Charpin and Gong on this topic.

After the improvement by Courtois and Meier of the algebraic attacks on stream ciphers and the introduction of the related notion of algebraic immunity, several constructions of infinite classes of Boolean functions with optimum algebraic immunity have been proposed. All of them gave functions whose algebraic degrees are high enough for resisting the Berlekamp-Massey attack and the recent Ronjom-Helleseth attack, but whose nonlinearities either achieve the worst possible value (given by Lobanov's bound) or are slightly superior to it. Hence, these functions do not allow resistance to fast correlation attacks. Moreover, they do not behave well with respect to fast algebraic attacks. In this paper, we study an infinite class of functions which achieve an optimum algebraic immunity. We prove that they have an optimum algebraic degree and a much better nonlinearity than all the previously obtained infinite classes of functions. We check that, at least for small values of the number of variables, the functions of this class have in fact a very good nonlinearity and also a good behavior against fast algebraic attacks.

To implement efficiently and securely good non-linear functions with a very large number of input variables is a challenge. Partially symmetric functions such as transitive functions are investigated to solve this issue. Known results on Boolean symmetric functions are extended both to transitive functions and to q-ary functions (on any set of qelements including finite fields GF(q) for any q). In a special case when the number of variables is n= pkwith pprime, an extension of Lucas' theorem provides new counting results and gives useful properties on the set of transitive functions. Results on balanced transitive q-ary functions are given. Implementation solutions are suggested based on q-ary multiple-valued decision diagrams and examples show simple implementations for these kind of symmetric functions. Applications include ciphers design and hash functions design but also search for improved covering radius of codes.

In this paper, we study the construction of (2t+1)-variable Boolean functions with maximum algebraic immunity, and we also analyze some other cryptographic properties of this kind of functions, such as nonlinearity, resilience. We first identify several classes of this kind of functions. Further, some necessary conditions of this kind of functions which also have higher nonlinearity are obtained. In this way, a modified construction method is proposed to possibly obtain (2t+1)-variable Boolean functions which have maximum algebraic immunity and higher nonlinearity, and a class of such functions is also obtained. Finally, we present a sufficient and necessary condition of (2t+1)-variable Boolean functions with maximum algebraic immunity which are also 1-resilient.

An analytic solution to the problem of a TE polarized plane electromagnetic wave scattering by two infinitely long conducting elliptic cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders. To compute the higher order terms of the scattered fields, the translation addition theorem for Mathieu functions is implemented to express the field scattered by one cylinder in terms of the elliptic coordinate system of the other cylinder in order to impose the boundary conditions. Scattered field coefficients of various scattered orders are obtained and written in matrix form. Numerical results are obtained for the scattered field in the far zone for different axial ratio, electrical separation distance and angles of incidence.

Algebraic attack has recently become an important tool in cryptanalysing different stream and block cipher systems. A Boolean function, when used in some cryptosystem, should be designed properly to resist this kind of attack. The cryptographic property of a Boolean function, that resists algebraic attack, is known as Algebraic Immunity ($\mathcal{AI}$). So far, the attempt in designing Boolean functions with required algebraic immunity was only ad-hoc, i.e., the functions were designed keeping in mind the other cryptographic criteria, and then it has been checked whether it can provide good algebraic immunity too. For the first time, in this paper, we present a construction method to generate Boolean functions on n variables with highest possible algebraic immunity ⌈n / 2⌉ . Such a function can be used in conjunction with (using direct sum) functions having other cryptographic properties. In a different direction we identify that functions, having low degree subfunctions, are weak in terms of algebraic immunity and analyse some existing constructions from this viewpoint.

This paper is an extended version of the lecture given at the Abel Conference 2002. It contains an exposition of several recent results in complex and real integral geometry inspired by Abel’s addition theorem and some applications to differential and functional equations.

Bent functions have maximal minimum distance to the set of affine functions. In other words, they achieve the maximal minimum distance to all the coordinate functions of affine monomials. In this paper we introduce a new class of bent functions which we call hyper-bent functions. Functions within this class achieve the maximal minimum distance to all the coordinate functions of all bijective monomials. We provide an explicit construction for such functions. We also extend our results to vectorial hyper-bent functions.

This paper addresses the problem of obtaining new construction methods for cryptographically significant Boolean functions. We show that for each positive integer m, there are infinitely many integers n (both odd and even), such that it is possible to construct n-variable, m-resilient functions having nonlinearity greater than 2n-1 -2[n/2]. Also we obtain better results than all published works on the construction of n-variable, m-resilient functions, including cases where the constructed functions have the maximum possible algebraic degree n - m - 1. Next we modify the Patterson-Wiedemann functions to construct balanced Boolean functions on n-variables having nonlinearity strictly greater than 2n-1 - 2n-1/2 for all odd n ≥ 15. In addition, we consider the properties strict avalanche criteria and propagation characteristics which are important for design of S-boxes in block ciphers and construct such functions with very high nonlinearity and algebraic degree.

Determining the weight of Boolean functions is of exponential complexity. By using combinatorial results, it is proved that from their algebraic normal form (ANF), it is possible to have polynomial time results on the weight, for some classes of functions. As a result, the structure of the majority functions MAJ2q-1 is given.

Functional decomposition—whether a function f(x) can be written as a composition of functions g(h(x)) in a nontrivial way—is an important primitive in symbolic computation systems. The problem of univariate polynomial decomposition was shown to have an efficient solution by Kozen and Landau [9]. Dickerson [5] and von zur Gathen [13] gave algorithms for certain multivariate cases. Zippel [15] showed how to decompose rational functions. In this paper, we address the issue of decomposition of algebraic functions. We show that the problem is related to univariate resultants in algebraic function fields, and in fact can be reformulated as a problem of resultant decomposition. We characterize all decompositions of a given algebraic function up to isomorphism, and give an exponential time algorithm for finding a nontrivial one if it exists. The algorithm involves genus calculations and constructing transcendental generators of fields of genus zero.

Trager (1984) recently gave a new algorithm for the indefinite integration of algebraic functions. His approach was “rational” in the sense that the only algebraic extension computed is the smallest one necessary to express the answer. We outline a generalization of this approach that allows us to integrate mixed elementary functions. Using only rational techniques, we are able to normalize the integrand, and to check a necessary condition for elementary integrability.