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In this paper, we study binomials having the form x r (a + x 3 (q - 1)) over the finite field F q 2 with q = 2 m , and determine all the r's and coefficients a's making them permutations. For even m or odd m with 3 ∤ m , we prove that the characterization is necessary and sufficient. For the case of odd m and 3 ∣ m , we prove that the corresponding sufficient condition is also necessary for almost all r's. Finally we obtain that the proportion of r's we cannot prove the necessity is only about 1 40 . [ABSTRACT FROM AUTHOR]

In Eurocrypt’18, Cid et al. proposed a new cryptanalysis tool called Boomerang Connectivity Table (BCT), to evaluate S-boxes of block ciphers. Later, Boura and Canteaut further investigated the new parameter Boomerang uniformity for cryptographic S-boxes. It is of great interest to find new S-boxes with low Boomerang uniformity for even dimensions. In this paper, we prove that a class of permutation quadrinomials over $\mathbb {F}_{2^{2m}}$ with $m$ odd has Boomerang uniformity four, which gives the fifth class of such kind of permutation polynomials. Further, the occurrences of 0 and 4 in the BCTs of the investigated permutation polynomials are also completely determined. [ABSTRACT FROM AUTHOR]

This paper studies the permutation behavior of the polynomial f (x) = x + a 1 x q (q − 1) + 1 + a 2 x 2 (q − 1) + 1 in F q 2 [ x ] for odd q, and finds a set of the coefficient pairs (a 1 , a 2) that leads f (x) to be a permutation of F q 2 . We transform the problem of proving that f is a permutation into determining the number of solutions to some low-degree equations in the unit circle of F q 2 . [ABSTRACT FROM AUTHOR]

In this paper, a family of non-monomial permutations over the finite field F2n with differential uniformity at most 6 is proposed, where n is a positive integer. The algebraic degree of these functions is also determined. [ABSTRACT FROM AUTHOR]

Are there other Boolean functions having two distinct Walsh coefficients except affine Boolean functions and maximal nonlinear (i.e. bent) Boolean functions? This paper proves that all Boolean functions with exactly two distinct Walsh coefficients are just the two known classes of affine and bent Boolean functions and the Boolean functions obtained by modifying the value of affine or bent Boolean functions at x = 0. [ABSTRACT FROM AUTHOR]