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

We study the essential properties of weakly regular p-ary bent functions of ℓ-form, where a p-ary function is from Fp_m to F_p. We observe that most of studies on a weakly 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.