On the nonlinearity of idempotent quadratic functions and the weight distribution of subcodes of Reed-Muller codes

International audienceThe Walsh transform \hat{Q} of a quadratic function Q : F2^n → F2 satisfies |\hat{Q(b)}| ∈ {0, 2 n+s 2 } for all b ∈ F_{2^n} , where 0 ≤ s ≤ n − 1 is an integer depending on Q. In this article, we investigate two classes of such quadratic Boolean functions which attracted a lot of research interest. For arbitrary integers n we determine the distribution of the parameter s for both of the classes, C1 = {Q(x) = Tr_n(\sum^{(n−1)/2}_{ i=1} a_ix^{2^i +1}) : a_i ∈ F2}, and the larger class C2, defined for even n as C2 = {Q(x) = Tr_n(^{(n/2)−1}_ { i=1} a_ix^{2^i +1}) + Tr_n/2 (a_{n/2} x^{2^n/2 +1}) : a_i ∈ F2}. Our results have two main consequences. We obtain the distribution of the non-linearity for the rotation symmetric quadratic Boolean functions, which have been attracting considerable attention recently. We also present the complete weight distribution of the corresponding subcodes of the second order Reed-Muller codes

Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

The Walsh transform QˆQ^ of a quadratic function Q:Fpn→FpQ:Fpn→Fp satisfies |Qˆ(b)|∈{0,pn+s2}|Q^(b)|∈{0,pn+s2} for all b∈Fpnb∈Fpn , where 0≤s≤n−10≤s≤n−1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class C1C1 is defined for arbitrary n as C1={Q(x)=Trn(∑⌊(n−1)/2⌋i=1aix2i+1):ai∈F2}C1={Q(x)=Trn(∑i=1⌊(n−1)/2⌋aix2i+1):ai∈F2} , and the larger class C2C2 is defined for even n as C2={Q(x)=Trn(∑(n/2)−1i=1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2}C2={Q(x)=Trn(∑i=1(n/2)−1aix2i+1)+Trn/2(an/2x2n/2+1):ai∈F2} . For an odd prime p, the subclass DD of all p-ary quadratic functions is defined as D={Q(x)=Trn(∑⌊n/2⌋i=0aixpi+1):ai∈Fp}D={Q(x)=Trn(∑i=0⌊n/2⌋aixpi+1):ai∈Fp} . We determine the generating function for the distribution of the parameter s for C1,C2C1,C2 and DD . As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case p>2p>2 , the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to C1C1 and C2C2 in terms of a generating function

The Carlitz rank of permutations of finite fields: a survey

L. Carlitz proved that any permutation polynomial f of a finite field Fq is a composition of linear polynomials and the monomials xq−2. This result motivated the study of Carlitz rank of f, which is defined in 2009 to be the minimum number of inversions xq−2, needed to obtain f, by E. Aksoy et al. We give a survey of results obtained so far on natural questions related to this concept and indicate a variety of applications, which emerged recently

On the inversive pseudorandom number generator

The inversive generator was introduced by J. Eichenauer and J. Lehn in 1986. A large number of papers on this generator have appeared in the last three decades, some investigating its properties, some generalizing it. It has been shown that the generated sequence and its variants behave very favorably with respect to most measures of randomness. In this survey article we present a comprehensive overview of results on the inversive generator, its generalizations and variants. As regards to recent work, our emphasis is on a particular generalization, focusing on the underlying permutation P(x) = ax(p-2) + b of F-p

Enumeration of quadratic functions with prescribed Walsh spectrum

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