Representations of Generalized Self-Shrunken Sequences

Sara D. Cardell, Joan-Josep Climent, Amparo Fúster-Sabater and Verónica Requena
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Sara D. Cardell: Instituto de Matemática, Estatística e Computação Científica, UNICAMP, 13083-859 Campinas-SP, Brazil
Joan-Josep Climent: Departament de Matemàtiques, Universitat d’Alacant, E-03690 Alacant, Spain
Amparo Fúster-Sabater: Instituto de Tecnologías Físicas y de la Información, C.S.I.C., E-28006 Madrid, Spain
Verónica Requena: Departament de Matemàtiques, Universitat d’Alacant, E-03690 Alacant, Spain

Mathematics, 2020, vol. 8, issue 6, 1-26

Abstract: Output sequences of the cryptographic pseudo-random number generator, known as the generalized self-shrinking generator, are obtained self-decimating Pseudo-Noise (PN)-sequences with shifted versions of themselves. In this paper, we present three different representations of this family of sequences. Two of them, the p and G -representations, are based on the parameters p and G corresponding to shifts and binary vectors, respectively, used to compute the shifted versions of the original PN-sequence. In addition, such sequences can be also computed as the binary sum of diagonals of the Sierpinski’s triangle. This is called the B -representation. Characteristics and generalities of the three representations are analyzed in detail. Under such representations, we determine some properties of these cryptographic sequences. Furthermore, these sequences form a family that has a group structure with the bit-wise XOR operation.

Keywords: generalized self-shrinking generator; PN-sequence; binomial sequence; additive group; coset (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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