Representations of Generalized Self-Shrunken Sequences

• Main Authors: Sara D. Cardell, Joan-Josep Climent, Amparo Fúster-Sabater, Verónica Requena
• Format: Article
• Language: English
• Published: MDPI AG 2020-06-01
• Series: Mathematics
• Subjects: generalized self-shrinking generator; PN-sequence; binomial sequence; additive group; coset
• Online Access: https://www.mdpi.com/2227-7390/8/6/1006

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 <i>p</i> and <i>G</i>-representations, are based on the parameters <i>p</i> and <i>G</i> 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 <i>B</i>-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.