Linear complexity profiles and continued fractions

Abstract

The linear complexity, L(sn), of a sequence sn is defined as the length of the shortest linear feedback shift-register (LFSR) that can generate the sequence. The linear complexity profile, Lsn = L1L2….Ln, of sn (where Li = L(si), 1 ≤ i ≤ n, denotes the linear complexity of first i digits of sn) provides better insight into the complexity of an individual sequence. By the increment sequence Δsn = Δ1 Δ2… Δm in a linear complexity profile, L1L2… Ln, we mean the subsequence of positive numbers in the sequence L1 (L2 - L1)… (Ln - Ln-1). For example, if L1… L5 = 0 2 2 2 3, its increment sequence is Δgs = Δ1Δ2 = 21. If we associate a sequence sn over F with an element S(z) in the field of Laurent series over F in the following way. (Formula Presented.) S(z) cm then be written as (Formula Presented.) where ai(z) ∈ F[z], the ring of polynomials in z over F, for all i ≥ 0. It will be shown that, for a sequence sn, the increment sequence Δsn of the linear complexity profile of sn is as follows. (1) If 2 · Σi=1k deg(ai(z)) - deg(ak(z)) ≤ n, then Δsn = deg(a1(z)) deg(a2(z)) ···, deg(ak(z)). (2) If 2 · Σi=1k deg(ai(z)) - deg(ak(z)) > n, then Δsn = deg(a1(z)) deg(a2(z)) ···, deg(ak(z)), where k1 = max{j : 2 · Σi=1j, deg(ai(z) - deg(aj(z)) ≤ n}.