On the construction of new bent functions from the max-weight and min-weight functions of old bent functions

Abstract

Given a bent function $$f(\varvec{x})$$ of n variables, its max-weight and min-weight functions are introduced as the Boolean functions $${f}^{+}(\varvec{x})$$ and $${f}^{-}(\varvec{x})$$ whose supports are the sets $$\{\varvec{a} \in {\mathbb {F}}_{2}^{n} | w(f \oplus l_{\varvec{a}}) = 2^{n-1}+2^{\frac{n}{2}-1}\}$$ and $$\{\varvec{a} \in {\mathbb {F}}_{2}^{n} | w(f \oplus l_{\varvec{a}}) = 2^{n-1}-2^{\frac{n}{2}-1}\}$$ respectively, where $$w(f \oplus l_{\varvec{a}})$$ denotes the Hamming weight of the Boolean function $$f(\varvec{x}) \oplus l_{\varvec{a}}(\varvec{x})$$ and $$l_{\varvec{a}}(\varvec{x})$$ is the linear function defined by $$\varvec{a} \in {\mathbb {F}}_{2}^{n}$$ . $${f}^{+}(\varvec{x})$$ and $${f}^{-}(\varvec{x})$$ are proved to be bent functions. Furthermore, combining the 4 minterms of 2 variables with the max-weight or min-weight functions of a 4-tuple $$(f_{0}(\varvec{x}), f_{1}(\varvec{x}), f_{2}(\varvec{x}), f_{3}(\varvec{x}))$$ of bent functions of n variables such that $$f_{0}(\varvec{x}) \oplus f_{1}(\varvec{x}) \oplus f_{2}(\varvec{x}) \oplus f_{3}(\varvec{x}) = 1$$ , a bent function of $$n+2$$ variables is obtained. A family of 4-tuples of bent functions satisfying the above condition is introduced, and finally, the number of bent functions we can construct using the method introduced in this paper are obtained. Also, our construction is compared with other constructions of bent functions.