Construction and equivalence for generalized boolean functions
Ayça Çeşmelioğlu, Wilfried Meidl · Cryptography and Communications · 2025
Abstract Recently in Çeşmelioğlu, Meidl (Adv. Math. Commun., 18, 2024), the study of EA-equivalence and CCZ-equivalence for functions from $${\mathbb {V}}_n^{(p)}$$ V n ( p ) to the cyclic group $${\mathbb {Z}}_{p^k}$$ Z p k has been initiated, where $$ {\mathbb {V}}_n^{(p)}$$ V n ( p ) denotes an n-dimensional vector space over $${\mathbb {F}}_p$$ F p . Amongst others it has been shown that there exist functions from $${\mathbb {V}}_n^{(2)}$$ V n ( 2 ) to $${\mathbb {Z}}_4$$ Z 4 which are CCZ-equivalent but not EA-equivalent. We extend these results to larger classes of functions from $${\mathbb {V}}_n^{(p)}$$ V n ( p ) to $${\mathbb {Z}}_{p^k}$$ Z p k . We then discuss constructions of generalized bent functions from $${\mathbb {V}}_n^{(p)}$$ V n ( p ) to $${\mathbb {Z}}_{p^k}$$ Z p k , p odd or $$p=2$$ p = 2 and n is even, which correspond to large affine spaces of bent functions. In particular we employ versions of the direct sum, the semi-direct sum and of a recent secondary bent function construction in Wang et. al., (IEEE Trans. Inform. Theory 69, 2023), to generate large affine spaces of bent functions. Finally we present a solution for constructing generalized bent functions from $${\mathbb {V}}_n^{(2)}$$ V n ( 2 ) to $${\mathbb {Z}}_{2^k}$$ Z 2 k , n odd, from arbitrary generalized bent functions from $${\mathbb {V}}_{n-1}^{(2)}$$ V n - 1 ( 2 ) to $${\mathbb {Z}}_{2^{k-1}}$$ Z 2 k - 1 .