Effective Construction of a Class of Bent Quadratic Boolean Functions

In this paper, we consider the characterization of the bentness of quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1} Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}}) ,$ where $n=me$, $m$ is even and $c_i\in GF(2^e)$. For a general $m$, it is difficult to determine the bentness of these functions. We present the bentness of quadratic Boolean function for two cases: $m=2^vp^r$ and $m=2^vpq$, where $p$ and $q$ are two distinct primes. Further, we give the enumeration of quadratic bent functions for the case $m=2^vpq$

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

A new class of semi-bent quadratic Boolean functions

In this paper, we present a new class of semi-bent quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\lfloor\frac{m-1}{2}\rfloor}Tr^n_1(c_ix^{1+4^{i}})$ $~(c_i\in \mathbb{F}_4$,$n=2m)$. We first characterize the semi-bentness of these quadratic Boolean functions. There exists semi-bent functions only when $m$ is odd. For the case: $m=p^r$, where $p$ is an odd prime with some conditions, we enumerate the semi-bent functions. Further, we give a simple characterization of semi-bentness for these functions with linear properties of $c_i$. In particular, for a special case of $p$, any quadratic Boolean function $f(x)=\sum_{i=1}^{\frac{p-1}{2}}Tr^{2p}_1(c_ix^{1+4^{i}})$ over $\mathbb{F}_{2^{2p}}$ is a semi-bent function

New Quadratic Bent Functions in Polynomial Forms with Coefficients in Extension Fields

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form $f(x)=\sum_{i=1}^{\frac{n}{2}-1}Tr^n_1(c_ix^{1+2^i})+ Tr_1^{n/2}(c_{n/2}x^{1+2^{n/2}})$, where $c_i\in GF(2^n)$ for $1\leq i \leq \frac{n}{2}-1$ and $c_{n/2}\in GF(2^{n/2})$. The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Further, we consider a subclass of quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1}Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}})$ , where $c_i\in GF(2^e)$, $n=em$ and $m$ is even. The bentness of these functions are characterized and some methods for constructing new quadratic bent functions are given. Finally, for a special case: $m=2^{v_0}p^r$ and $gcd(e,p-1)=1$, we present the enumeration of quadratic bent functions

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