Nyberg construction nonlinear transforms based on all isomorphic representations of the Galois field GF(512)

Abstract

The paper deals with questions of increasing the efficiency of block symmetric cryptographic algorithms used in modern telecommunications systems. One of the most important elements of any modern symmetric block cryptographic algorithm is its nonlinear transform or S -box, which largely determines its cryptographic features and performance. Therefore the task of development of a large set of cryptographic S -boxes with a high level of quality is the key to solving the problem of increasing the efficiency of modern cryptographic algorithms. The basis of one of the most common block symmetric cryptographic algorithms AES/Rijndael is the nonlinear transform of Nyberg design of length N = 256. This design allows construction of a small set of nonlinear transforms which have a high level of cryptographic quality. However, the development of modern computer technology dictates not only the task of building of S -boxes of greater length, as well as finding ways to increase the cardinality of sets of available high quality non-linear transforms. In this paper a class of S -boxes of Nyberg design with cardinality J = 392 and the length N = 512 over all the isomorphic representations of the field GF(512) is built, which allows improvement of the efficiency of cryptographic nonlinear transforms, as well as to increase the cardinality of their sets. The dynamics of improvement of the cryptographic quality of S -boxes of Nyberg design with increasing of their length are investigated. It was found that increasing of the length of nonlinear transforms of Nyberg design gives the rapid decline of correlation between vectors of output and input of S -box as well as significant growth of its distance of nonlinearity. It is shown that great length S -boxes of Nyberg design obtains such distance of nonlinearity, that it becomes almost equivalent to the distance of the nonlinearity of Boolean bent-functions.