Further results on differentially 4-uniform permutations over $\mathbb{F}_{2^{2m} } $

Abstract

In this paper, we present several new constructions of differentially 4-uniform permutations over F 2 2m by modifying the values of the inverse function on some subsets of F 2 2m . The resulted differentially 4-uniform permutations have high nonlinearities and algebraic degrees, which provide more choices for the design of crytographic substitution boxes.

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

“…Qu et al [QTTL13,QTLG16], Peng et al [PT16] and Tang et al [TCT15] proposed several families of differentially 4-uniform permutations with optimal algebraic degree from the inverse function by applying the powerful switching method. Later, Zha et al [ZHS14,ZHSS15] presented some more families of differentially 4-uniform permutations with optimal algebraic degree by applying affine transformations on the elements of some subfields of the inverse function. In [CTTL14], Carlet et al built a family of differentially 4-uniform permutations with optimal algebraic degree by concatenating two functions from F 2 n−1 to F 2 n for even n ≥ 6.…”

Two classes of differentially 4-uniform permutations over $ \mathbb{F}_{2^{n}} $ with $ n $ even

“…The nonlinearity bound of these functions can be obtained by using Theorem 3.5 in [15]. By comparing our newly constructed functions with previous constructions in [17][18][19]25], we found that the differential spectrum or the nonlinearity of the functions in Table 1 are different from that in those papers (see Table 3 in [17], Tables 2 and 3 in [18], Tables III and IV in [19] and Table 2 in [25]), which implies that the functions in Table 1 are CCZ-inequivalent to all the functions in those papers. Unfortunately, our numerical results show that the construction in Theorem 3.2 does not provide a function with nonlinearity better than Construction 5.5 in [15], although our construction can get more differentially 4-uniform permutations polynomials.…”

“…Compared to the numerical results in [15, 17-19, 21, 23-25], Table 2 presents new results on nonlinearities and differential spectra for n = 12, 6, 8. This implies that the differentially 4-uniform permutations defined in Theorems 4.4 and 4.8 are CCZ-inequivalent to all the functions in [15, 17-19, 21, 23-25] (see Table 2 in [15], Tables 2, 3 and 4 in [17], Tables 2 and 3 in [18],Tables II, III and IV in [19], Table 3 in [21], Table A.1 in [23], Tables 1 and 2 in [24] and Tables 1 and 3 in [25]).…”

“…Due to the best known nonlinearity and maximum algebraic degree of the inverse function over F 2 n for even n, a lot of differentially 4-uniform permutations have been constructed by modifying the values of the inverse function on some special subsets of F 2 n . A non-exhaustive list of references dealing with the known differentially 4-uniform functions derived from the inverse functions is [5,12,[16][17][18][19][20][21][23][24][25].…”

Characterizing differential support of vectorial Boolean functions using the Walsh transform

“…There are very few known APN functions, and they may have potential drawbacks. In recent years, increasing attention has been given to differentially 4/6-uniform permutations (see [2][3][4][5] and references therein). Another key property of vectorial functions is the nonlinearity, which is characterized by the Walsh transforms of the functions.…”