REGULAR METHOD FOR SYNTHESIS OF BASIC BENT-SQUARES OF RANDOM ORDER

Abstract

The paper is devoted to the class construction of the most non-linear Boolean bent-functions of any length N = 2^k (k = 2, 4, 6…), on the basis of their spectral representation – Agievich bent squares. These perfect algebraic constructions are used as a basis to build many new cryptographic primitives, such as generators of pseudo-random key sequences, crypto graphic S-boxes, etc. Bent-functions also find their application in the construction of C-codes in the systems with code division multiple access (CDMA) to provide the lowest possible value of Peak-to-Average Power Ratio (PAPR) k = 1, as well as for the construction of error-correcting codes and systems of orthogonal biphasic signals. All the numerous applications of bent-functions relate to the theory of their synthesis. However, regular methods for complete class synthesis of bent-functions of any length N = 2^k are currently unknown. The paper proposes a regular synthesis method for the basic Agievich bent squares of any order n, based on a regular operator of dyadic shift. Classification for a complete set of spectral vectors of lengths (l = 8, 16, …) based on a criterion of the maximum absolute value and set of absolute values of spectral components has been carried out in the paper. It has been shown that any spectral vector can be a basis for building bent squares. Results of the synthesis for the Agievich bent squares of order n = 8 have been generalized and it has been revealed that there are only 3 basic bent squares for this order, while the other 5 can be obtained with help the operation of step-cyclic shift. All the basic bent squares of order n = 16 have been synthesized that allows to construct the bent-functions of length N = 256. The obtained basic bent squares can be used either for direct synthesis of bent-functions and their practical application or for further research in order to synthesize new structures of bent squares of orders n = 16, 32, 64, …