Cryptographically strong permutations from the butterfly structure

Boomerang connectivity table is a new tool to characterize the vulnerability of cryptographic functions against boomerang attacks. Consequently, a cryptographic function is desired to have boomerang uniformity as low as its differential uniformity. Based on generalized butterfly structures recently introduced by Canteaut, Duval and Perrin, this paper presents infinite families of permutations of ${\mathbb {F}}_{2^{2n}}$ for a positive odd integer n, which have the best known nonlinearity and boomerang uniformity 4. Both open and closed butterfly structures are considered. The open butterflies, according to experimental results, appear not to produce permutations with boomerang uniformity 4. On the other hand, from the closed butterflies we derive a condition on coefficients $\alpha , \beta \in {\mathbb {F}}_{2^n}$ such that the functions $$\begin{aligned} V_i(x,y) := (R_i(x,y), R_i(y,x)), \end{aligned}$$ where $R_i(x,y)=(x+\alpha y)^{2^i+1}+\beta y^{2^i+1}$ and $\gcd (i,n)=1$, permute ${{\mathbb {F}}}_{2^n}^2$ and have boomerang uniformity 4. In addition, experimental results for $n=3, 5$ indicate that the proposed condition seems to cover all such permutations $V_i(x,y)$ with boomerang uniformity 4.acceptedVersio

On the Generalization of Butterfly Structure

Butterfly structure was proposed in CRYPTO 2016 [PUB16], and it cangenerate permutations over F22n from power permutations over F2n for odd n. Afterthat, a generalized butterfly structure was proposed in IEEE IT [CDP17], which cangenerate permutations over F22n from any permutation over F2n . There is also anothergeneralization which was given in [FFW17]. Up to now, three constructions based onbutterfly structure and Gold type permutations are proposed. In the present paper,we give a construction which contains the three previous constructions as special casesand also generates new permutations with good cryptographic properties. Moreover,we give a characterization of the number of solutions of a special system of linearequations in a more general way, which is useful to investigate the cryptographicproperties of quadratic functions obtained with butterfly construction based on Goldexponents