More permutation polynomials with differential uniformity six

Differential analysis is one of the most important attacks threatening iterated block ciphers. To resist against differential attacks [1], the multioutput Boolean functions used in designing Sboxes should have low differential uniformity. For a positive integer n, the lowest differential uniformity of the functions from the finite field F2n (for a prime power q, Fq denotes the finite field with q elements, and Fq denotes Fq \ {0}) to itself is 2, and these functions are called almost perfect nonlinear (APN). APN functions have been intensively studied in the last decades and researchers have made many interesting observations about them (the reader is referred to [2] and references therein). While the cubic map is an obvious APN permutation for odd dimension n, the existence of APN permutations for even dimension n has been a long-standing question. Very recently, the first example of APN permutation over F26 was found in [3], but the question on the existence of APN permutations in even dimension n > 6 still remains open. From the cryptographic point of view, permutations with low differential uniformity are therefore of great interest [4].