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Title:
New Quadratic Bent Functions in Polynomial Forms with Coefficients in Extension Fields
Authors: Chunming Tang, Yanfeng Qi, Maozhi Xu
Abstract: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.
ePrint: https://eprint.iacr.org/2013/405
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