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A Group Law on the Projective Plane with Applications in Public Key Cryptography

Representations of Generalized Self-Shrunken Sequences

Instituto de Matemática, Estatística e Computação Científica, UNICAMP, 13083-859 Campinas-SP, Brazil
Departament de Matemàtiques, Universitat d’Alacant, E-03690 Alacant, Spain
Instituto de Tecnologías Físicas y de la Información, C.S.I.C., E-28006 Madrid, Spain
Author to whom correspondence should be addressed.
Mathematics 2020, 8(6), 1006;
Received: 26 May 2020 / Revised: 12 June 2020 / Accepted: 15 June 2020 / Published: 19 June 2020
(This article belongs to the Special Issue Mathematics Cryptography and Information Security)
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. View Full-Text
Keywords: generalized self-shrinking generator; PN-sequence; binomial sequence; additive group; coset generalized self-shrinking generator; PN-sequence; binomial sequence; additive group; coset
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MDPI and ACS Style

Cardell, S.D.; Climent, J.-J.; Fúster-Sabater, A.; Requena, V. Representations of Generalized Self-Shrunken Sequences. Mathematics 2020, 8, 1006.

AMA Style

Cardell SD, Climent J-J, Fúster-Sabater A, Requena V. Representations of Generalized Self-Shrunken Sequences. Mathematics. 2020; 8(6):1006.

Chicago/Turabian Style

Cardell, Sara D., Joan-Josep Climent, Amparo Fúster-Sabater, and Verónica Requena. 2020. "Representations of Generalized Self-Shrunken Sequences" Mathematics 8, no. 6: 1006.

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