EA-equivalence of Section 3 binomial bent functions to the general class in Section 4

Determine whether the ternary binomial bent functions f(x) = Tr_n(a x^{2(3^k+1)} + a^{-1} x^{(3^k+1)^2}) constructed for n = 4k with k ≡ 3 (mod 4) and a ∈ F_{3^4} satisfying a^4 + a − 1 = 0 (Section 3, Theorem 1) are extended-affine equivalent to the general binomial bent functions f(x) = Tr_n(a1 x^{2(3^k+1)} + a2 x^{(3^k+1)^2}) defined for n = 4k with a1 a nonsquare in F_{3^{4k}} and a2 given by a2 = ± I^k·a1^{(3^k+1)/2}·((−1)^k·a1^{((3^k−1)(3^{2k}+1)/4)} + a1^{−((3^k−1)(3^{2k}+1)/4)}), where I is a primitive 4th root of unity in F_{3^{4k}} (Section 4, Theorem 4).

Background

The paper constructs two classes of ternary binomial bent functions within the completed Maiorana–McFarland class. Section 3 presents a specific binomial family for n = 4k with k ≡ 3 (mod 4), given in univariate form by f(x) = Tr_n(a x^{2(3^k+1)} + a^{-1} x^{(3^k+1)^2}), where a ∈ F_{3^4} satisfies a^4 + a − 1 = 0. Section 4 develops a more general binomial family for n = 4k of the form f(x) = Tr_n(a1 x^{2(3^k+1)} + a2 x^{(3^k+1)^2}), where a1 is a nonsquare in F_{3^{4k}} and a2 is defined via a specific formula involving a1 and a primitive 4th root of unity I.

Extended-affine (EA) equivalence is a structural notion that partitions functions into equivalence classes under affine permutations and affine additions. The authors report computational evidence that the Section 3 binomial functions are EA-equivalent to instances from the more general class in Section 4 for small dimensions, but they do not have a general proof. Establishing this equivalence (or disproving it) would clarify the relationship between these two constructions and potentially unify them within a single EA-equivalence class.