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Two infinite classes of rotation symmetric bent functions with simple representation

Chunming Tang,Yanfeng Qi,Zhengchun Zhou,Cuiling Fan

Published 2015 in Applicable Algebra in Engineering, Communication and Computing

ABSTRACT

In the literature, few n-variable rotation symmetric bent functions have been constructed. In this paper, we present two infinite classes of rotation symmetric bent functions on $${\mathbb {F}}_2^{n}$$F2n of the two forms:(i)$$f(x)=\sum _{i=0}^{m-1}x_ix_{i+m} + {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})$$f(x)=∑i=0m-1xixi+m+γ(x0+xm,…,xm-1+x2m-1),(ii)$$f_t(x)= \sum _{i=0}^{n-1}(x_ix_{i+t}x_{i+m} +x_{i}x_{i+t})+ \sum _{i=0}^{m-1}x_ix_{i+m}+ {\upgamma }(x_0+x_m,\ldots , x_{m-1}+x_{2m-1})$$ft(x)=∑i=0n-1(xixi+txi+m+xixi+t)+∑i=0m-1xixi+m+γ(x0+xm,…,xm-1+x2m-1), where $$n=2m$$n=2m, $${\upgamma }(X_0,X_1,\ldots , X_{m-1})$$γ(X0,X1,…,Xm-1) is any rotation symmetric polynomial, and $$m/\textit{gcd}(m,t)$$m/gcd(m,t) is odd. The class (i) of rotation symmetric bent functions has algebraic degree ranging from 2 to m and the other class (ii) has algebraic degree ranging from 3 to m. Moreover, the two classes of rotation symmetric bent functions are disjoint.

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