Representations of Generalized Self-Shrunken Sequences

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.

Review of the Lineal Complexity Calculation through Binomial Decomposition-Based Algorithms

“…This algorithm is based on the B-representation (or Binomial representation) [17] of a binary sequence {s n } n≥0 with length l = 2 m , m being a non-negative integer. Via the B-representation, the parameter LC of such a sequence is analyzed and computed.…”

“…,h 2 m −1 ). As {s n } n≥0 is a binary sequence of length l = 2 m and given the (2 m × 2 m ) binomial matrix H m , we compute the vector c c c whose 2 m components are the coefficients c i by means of the equation (see [17] (Section 3.2)):…”

“…Despite it being based on our previous design of the folding algorithm [16], in this work we complete the available knowledge on such an algorithm providing a mathematical proof of its behaviour and correctness. The matrix binomial decomposition algorithm is another novelty of this article, which is based on a recent representation of the generalized self-shrunken sequences [17]. Finally, after completing the gaps on the algorithm definitions, the last contribution of our work is the comparison among all the previous algorithms and the discussion about their different use-cases.…”

An Efficient Algorithm to Compute the Linear Complexity of Binary Sequences

“…In fact, our proposal is an algorithm that computes the linear complexity of sequences with a period of a power of two. The technique herein presented uses Hadamard matrices [25] and the B-representation of sequences proposed in [26,27]. Our algorithm was much more efficient than the other ones proposed in the literature [26,[28][29][30], and the required amount of sequence to perform this computation was also more realistic.…”

Folding-BSD Algorithm for Binary Sequence Decomposition

“…The LC of PN-sequences equals L, while that of the GSS-sequences is near to 2 L−1 [22,23]. This property is desirable because such sequences exhibit a great LC with very low resources.…”