Cryptographically Strong Permutations from the Butterfly Structure
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In this paper, we present infinite families of permutations of F_2^2n with high nonlinearity and boomerang uniformity 4 from generalized butterfly structures. Both open and closed butterfly structures are considered. It appears, according to experiment results, that open butterflies do not produce permutation with boomerang uniformity 4. For the closed butterflies, we propose the condition on coefficients α, β∈F_2^n such that the functions V_i := (R_i(x,y), R_i(y,x)) with R_i(x,y)=(x+α y)^2^i+1+β y^2^i+1 are permutations of F_2^n^2 with boomerang uniformity 4, where n≥ 1 is an odd integer and (i, n)=1. The main result in this paper consists of two major parts: the permutation property of V_i is investigated in terms of the univariate form, and the boomerang uniformity is examined in terms of the original bivariate form. In addition, experiment results for n=3, 5 indicates that the proposed condition seems to cover all coefficients α, β∈F_2^n that produce permutations V_i with boomerang uniformity 4.
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