Abstract Recently, Baudrin et al. analyzed a special case of Wagner’s commutative diagram cryptanalysis, referred to as commutative cryptanalysis. For a family $$(E_k)_k$$ ( E k ) k of permutations on a finite vector space G, commutative cryptanalysis exploits the existence of affine permutations $$A,B :G \rightarrow G$$ A , B : G → G , $$I \notin \{A,B\}$$ I ∉ { A , B } such that $$E_k \circ A (x) = B \circ E_k(x)$$ E k ∘ A ( x ) = B ∘ E k ( x ) holds with high probability, taken over inputs x, for a significantly large set of weak keys k. Several attacks against symmetric cryptographic primitives can be formulated within the framework of commutative cryptanalysis, most importantly differential attacks, as well as rotational and rotational-differential attacks. Besides, the notion of c-differentials on S-boxes can be analyzed as a special case within this framework. We discuss the relations between a general notion of commutative cryptanalysis, with A and B being arbitrary functions over a finite Abelian group, and differential cryptanalysis, both from the view of conducting an attack on a symmetric cryptographic primitive, as well as from the view of a theoretical study of cryptographic S-boxes.

In this paper we define a new (output) multiplicative differential, and the corresponding c-differential uniformity. With this new concept, even for characteristic 2, there are perfect c-nonlinear (PcN) functions. We first characterize the c-differential uniformity of a function in terms of its Walsh transform. We further look at some of the known perfect nonlinear (PN) functions and show that only one remains a PcN function, under a different condition on the parameters. In fact, the p-ary Gold PN function increases its c-differential uniformity significantly, under some conditions on the parameters. We then precisely characterize the c-differential uniformity of the inverse function (in any dimension and characteristic), relevant for the Rijndael (and Advanced Encryption Standard) block cipher.

We exhibit an infinite class of almost perfect nonlinear quadratic polynomials from F2nto F2n(n ges 12, n divisible by 3 but not by 9). We prove that these functions are EA-inequivalent to any power function and that they are CCZ-inequivalent to any Gold function. In a forthcoming full paper, we shall also prove that at least some of these functions are CCZ-inequivalent to any Kasami function

Suppose that n=2k is even. We study the cross-correlation function between two m-sequences for Niho type decimations d=(2/sup k/-1)s+1. We develop a new technique to study the value distribution of these cross-correlation functions, which makes use of Dickson polynomials. As a first application, we derive here the distribution of the six-valued cross-correlation function for s=3 and odd k, up to a term which depends on Kloosterman sums. In addition, applying simpler methods, we prove a theorem providing Niho type decimations with four-valued cross-correlation functions and their distribution. We conjecture that the latter result actually covers all such decimations.