A family of binary sequences from interleaved construction and their cryptographic properties

Abstract

Families of pseudorandom sequences with low cross correlation have important applications in communications and cryptography. Among several known constructions of sequences with low cross correlations, interleaved constructions proposed by Gong uses two sequences of the same period with two-level autocorrelation. In this paper, we study the balance property and the cross correlation of interleaved sequences such that the base sequences may not have the same period, or they may not have two-level autocorrelation. In particular, we study the interleaved sequences of two Legendre sequences of periods p and q, respectively, where p and q are odd prime numbers.

Interleaved Sequences Over Finite Fields

“…We also give the number of zeros of every sequence in the constructed family in one period. The results can also be found in [35].…”

“…Now we consider the case that p ≥ 2t or p = t. First we consider the case p ≥ 2t and note that (38) is not possible in this case. Otherwise (38) and (35)…”

“…for the special case = p. However, the result is also applicable to the (generalized) Lee sequence over F with a prime power ( When τ > t the zero correlation of the interleaved sequence in Algorithm 6.5 can be destroyed. A discussion on the upper bound of this correlation for the case τ > t is given in [31]; see also [35]. Therefore, we restrict to 0 < |τ | ≤ t − 1 and modify Lemma 4.6 to the following new lemma.…”

“…Let a, b and c be three m−sequences and let e and f be two shift sequences. We have obtained a family of sequences with low correlation [35] which covers the case that the two base sequences are m−sequences originally given in [29]. We consider an obtained sequence s k ∈ S(a, b, e) to be our first base sequence and c to be our second base sequence.…”