On equivalence between APN polynomials in brivariate form and power functions

Constructing new APN functions and determining Carlet-Charpin-Zinoviev equivalence (CCZ-equivalence) between APN functions are important but difficult topics in the study of cryptographic functions. In this work, we present a way to construct APN functions that are CCZ-inequivalent to power functions. More precisely, we theoretically prove the functions in the form G (x, y) = (xy, xq'+1 + bxq' y + cxyq' + dyq'+1 + (d1 xq'+1 + d2 xq' y + d3 xyq' + d4yq'+1)2m/2) (q' = 2i) are CCZ-inequivalent to power functions under certain conditions, and we present APN instance over 𝔽28 from this construction. In 2021, Calderini [Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 25 (60), 79-105, (2021)] constructed an infinite family of APN functions over 𝔽22m (m is even) in the from F (x, y) = (xy, x2i+1 + x2i+m/2 y2m/2 + bxy2i + cy2i+1). By our theoretical CCZ-inequivalence result on G (x, y), we show the inequivalence between F and power functions under certain conditions.