Linear complexity of generalized sequences by comparison of PN-sequences

Abstract

Linear complexity is a much used metric of the security of any binary sequence with application in communication systems and cryptography. In this work, we propose a method of computing the linear complexity of a popular family of cryptographic sequences, the so-called generalized sequences. Such a family is generated by means of the irregular decimation of a single Pseudo Noise sequence (PN-sequence). The computation method is based on the comparison of the PN-sequence with shifted versions of itself. The concept of linear recurrence relationship and the rows of the Sierpinski triangle play a leading part in this computation.

Folding-BSD Algorithm for Binary Sequence Decomposition

“…Since the PN-sequence has 2 L−1 ones, the period of any generalized sequence will be 2 L−1 or divisors of this number, in any case, a power of 2. In addition, the LC of every GSS-sequence is upper-bounded by 2 L−1 − (L − 2) [22] (Theorem 2). Next, an illustrative example of a family of GSS-sequences is introduced.…”

“…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.…”

Representations of Generalized Self-Shrunken Sequences

“…This polynomial is known as the characteristic polynomial of the recurring sequence. A common metric of the security of a sequence for its possible cryptographic application is the linear complexity [39][40][41], denoted by LC. Roughly speaking, the parameter LC determines the portion of sequence we need in order to recover the whole sequence.…”

“…Relating to the linear complexity, in [39] Blackburn introduced an upper bound for the linear complexity of the self-shrinking generator. A generalization of this bound was introduced in [40] for the linear complexity of generalized sequences, that is, LC ≤ 2 L−1 − (L − 2). Furthermore, we know that for all generalized sequences, except for those with period 1 and 2, we have 2 L−2 ≤ LC (although there is no proof for this statement either).…”

Folding-BSD Algorithm for Binary Sequence Decomposition

“…Since the PN-sequence has 2 L−1 ones, the period of any generalized sequence will be 2 L−1 or divisors of this number, in any case, a power of 2. In addition, the LC of every GSS-sequence is upper-bounded by [22] (Theorem 2). Next, an illustrative example of a family of GSS-sequences is introduced.…”