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Enumeration of quadratic functions With Prescribed Walsh Spectrum

IEEE TRANSACTIONS ON INFORMATION THEORY 2014年第10期60卷 6669-6680页

作者： Meidl, Wilfried Roy, Sankhadip Topuzoglu, Alev Sabanci Univ Fac Engn & Nat Sci TR-34956 Istanbul Turkey

The Walsh transform (f) over cap of a quadratic function f : F-p(n) -> F-p satisfies vertical bar(f) over cap vertical bar epsilon{0, p(n+s/2)} for an integer 0 <= s <= n-1, depending on f. In this paper, quadratic functions of the form F-p,F-n(x) = Tr-n(Sigma(k)(i=0) a(i)x(pt+1)) are studied, with the restriction that a(i) is an element of F-p, 0 <= i <= k. Three methods for enumeration of such functions are presented when the value for s is prescribed. This paper extends earlier enumeration results significantly, for instance, the generating function for the counting function is obtained, when n is odd and relatively prime to p, or when n = 2m, for odd m and p = 2. The number of bent and semibent functions for various classes of n is also obtained.

关键词： quadratic boolean functions quadratic p-ary functions Walsh transform semi-bent functions plateaued functions discrete Fourier transform self-reciprocal polynomials

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quadratic functions with prescribed spectra

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DESIGNS CODES AND CRYPTOGRAPHY 2013年第1-3期66卷 257-273页

作者： Meidl, Wilfried Topuzoglu, Alev Sabanci Univ MDBF TR-34956 Istanbul Turkey

We study a class of quadratic p-ary functions from to , which are well-known to have plateaued Walsh spectrum;i.e., for each the Walsh transform satisfies for some integer 0 a parts per thousand currency sign s a parts per thousand currency sign n - 1. For various types of integers n, we determine possible values of s, construct with prescribed spectrum, and present enumeration results. Our work generalizes some of the earlier results, in characteristic two, of Khoo et. al. (Des Codes Cryptogr, 38, 279-295, 2006) and Charpin et al. (IEEE Trans Inf Theory 51, 4286-4298, 2005) on semi-bent functions, and of Fitzgerald (Finite Fields Appl 15, 69-81, 2009) on quadratic forms.

关键词： quadratic boolean functions quadratic p-ary functions Plateaued functions Semi-bent functions Self-reciprocal polynomials Linear complexity

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WEIGHT DISTRIBUTIONS OF COSETS OF 2-ERROR-CORRECTING BINARY BCH CODES, EXTENDED OR NOT

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IEEE TRANSACTIONS ON INFORMATION THEORY 1994年第5期40卷 1425-1442页

作者： CHARPIN, P INRIA Le Chesnay Fr

Let B be the binary two-error-correcting BCH code of length 2(m) - 1 and let ($) over cap B be the extended code of B. We give formal expressions of weight distributions of the cosets of the codes ($) over cap B only depending on m. We can then deduce the weight distributions of the cosets of B. When m is odd, it is well known that there are four distinct weight distributions for the cosets of B. So our main result is about the even case. In a recent paper, Camion, Courteau, and Montpetit observe that for the lengths 15, 63, and 255 there are eight distinct weight distributions. We prove that this property holds for the codes ($) over cap B and B for all even m.

关键词： BCH-CODES WEIGHT DISTRIBUTION OF COSETS quadratic boolean functions GROUP ALGEBRA

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