Enumeration of quadratic functions with prescribed Walsh spectrum

Meidl, Wilfried and Sankhadip, Roy and Topuzoğlu, Alev (2014) Enumeration of quadratic functions with prescribed Walsh spectrum. IEEE Transactions on Information Theory, 60 (10). pp. 6669-6680. ISSN 0018-9448 (Print) 1557-9654 (Online)

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Official URL: http://dx.doi.org/10.1109/TIT.2014.2341237


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.

Item Type:Article
Uncontrolled Keywords:Quadratic Boolean functions; quadratic p-ary functions; Walsh transform; semi-bent functions; plateaued functions; discrete Fourier transform; self-reciprocal polynomials
ID Code:25111
Deposited By:Alev Topuzoğlu
Deposited On:14 Nov 2014 22:32
Last Modified:14 Nov 2014 22:32

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