G-Perfect nonlinear functions

Abstract

Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both Z 2 -valued and Z a 2 -valued. Our final constructions demonstrate G-perfect nonlinear planar permutations (from Z a 2 to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.Keywords G-perfect nonlinear functions · Difference sets AMS Classifications 05B10 · 11T71 Background on cryptosystems and group action modificationsIn an r-round iterative block cipher such as the Data Encryption Standard (DES) [17] or the Advanced Encryption Standard (AES) [9,18] the ciphertext x r is obtained from a plaintext x 0 by r iterations of the round function f

Nonlinear functions and difference sets on group actions

“…We will also investigate the existence and constructions of G-perfect nonlinear functions and G-bent functions. Several known results in [2,6,10,17] are direct consequences of our results. …”

“…In this paper we will first give a characterization of G-perfect nonlinear functions on a G-set X by (G, H)-related difference families of X (see Theorem 2.6). As applications of this characterization, we obtain several known results in [2,6,17] as immediate consequences. When G is a finite abelian group, we introduce the concept of a normalized G-dual set (see Definition 3.6), and use it to characterize a G-difference set of X (see Theorem 4.3).…”

“…Applications of Theorem 2.6 will also be discussed, and several known results in [2,6,17] are obtained as direct consequences. Throughout the paper, the following notation will be used.…”

“…[4,6,7,[9][10][11]13,14,16,17]). They can be used to construct DES-like cryptosystems that are resistant to differential attacks.…”

“…They can also be used to construct generalized DES-like cryptosystems (where the XOR operation is replaced by a group action) that are resistant to modified differential attacks. For a summary of the background on cryptosystems and group action modifications, the reader is referred to [6]. Pott [14] mentioned that "It seems that in most applications (in particular in cryptography) people use nonlinear functions on finite fields.…”