Characterization of weakly regular p-ary bent functions of ℓ-form

초록

We study the essential properties ofweakly regular p-ary bent functions of l-form, where a pary function is fromFpm to Fp. We observe that most of studies on aweakly regular p-ary bent function f with f (0) = 0 of l-form always assume the gcd-condition: gcd( l- 1, p - 1) = 1. We first show that whenever considering weakly regular p-ary bent functions f with f (0) = 0 of l-form, we can drop the gcd-condition; using the gcd-condition, we also obtain a characterization of a weakly regular bent function of l-form. Furthermore, we find an additional characterization for weakly regular bent functions of l-form; we consider two cases m being even or odd. Let f be a weakly regular bent function of l-form preserving the zero element; then in the case that m is odd, we show that f satisfies gcd(l, p - 1) = 2. On the other hand, when m is even and f is also non-regular, we show that f satisfies gcd(l, p - 1) = 2 as well. In addition, we present two explicit families of regular bent functions of l-form in terms of the gcd-condition.

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