Two New Infinite Families of APN Functions in Trivariate Form
Kangquan Li, Nikolay Kaleyski
Abstract
We present two infinite families of APN functions in trivariate form over finite fields of the form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{2^{3m}}$ </tex-math></inline-formula> . We show that the functions from both families are permutations when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is odd, and are 3-to-1 functions when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> is even. In particular, our functions are AB permutations for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> odd. Furthermore, we observe that for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m = 3$ </tex-math></inline-formula> , i.e. for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{2^{9}}$ </tex-math></inline-formula> , the functions from our families are CCZ-equivalent to the two bijective sporadic APN instances discovered by Beierle and Leander. We thus generalize these sporadic instances into an infinite family of APN functions. We also perform an exhaustive computational search for quadratic APN functions with binary coefficients in trivariate form over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb F}_{2^{3m}}$ </tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m \le 5$ </tex-math></inline-formula> and report on the results.