Differentially 4-Uniform Permutations with the Best Known Nonlinearity from Butterflies

Recently, a new structure called butterfly introduced by Perrin et at. is attractive for that it has very good cryptographic properties: the differential uniformity is at most equal to 4 and algebraic degree is also very high when exponent e = 3. It is conjecture that the nonlinearity is also optimal for every odd k, which was proposed as a open problem. In this paper, we further study the butterfly structures and show that these structure with exponent e = 2 + 1 have also very good cryptographic properties. More importantly, we prove in theory the nonlinearity is optimal for every odd k, which completely solve the open problem. Finally, we study the butter structures with trivial coefficient and show these butterflies have also optimal nonlinearity. Furthermore, we show that the closed butterflies with trivial coefficient are bijective as well, which also can be used to serve as a cryptographic primitive.