The Asymptotic Normalized Linear Complexity of Multisequences

We show that the asymptotic linear complexity of a multisequence a in F_q^\infty that is I := liminf L_a(n)/n and S := limsup L_a(n)/n satisfy the inequalities M/(M+1) <= S <= 1 and M(1-S) <= I <= 1-S/M, if all M sequences have nonzero discrepancy infinitely often, and all pairs (I,S) satisfying these conditions are met by 2^{\aleph_0} multisequences a. This answers an Open Problem by Dai, Imamura, and Yang. Keywords: Linear complexity, multisequence, Battery Discharge Model, isometry.

💡 Research Summary

The paper investigates the asymptotic behaviour of the normalized linear complexity of multisequences over a finite field 𝔽_q. For a multisequence a = (a_{m,t}){1≤m≤M, t≥1} the linear complexity L_a(n) is defined as the smallest degree of a common denominator polynomial v(x) that simultaneously approximates the M formal power series G_m(x)=∑{t≥1} a_{m,t} x^{-t} up to order x^{-n}. Normalizing by n yields the ratio L_a(n)/n, which typically hovers around M/(M+1). The authors focus on the liminf I = lim inf L_a(n)/n and limsup S = lim sup L_a(n)/n, and they determine the exact region of admissible (I,S) pairs.

The technical core rests on two algorithmic tools. First, the multi‑Strict Continued Fraction Algorithm (mSCFA) introduced by Dai and Feng computes, for each position (m,n), a discrepancy δ(m,n)∈𝔽_q. If δ=0 the current rational approximation is exact; otherwise the algorithm updates the denominator degree. Second, the authors develop a probabilistic Battery‑Discharge Model (BDM) that abstracts the mSCFA dynamics. In the BDM, δ=0 occurs with probability 1/q and δ≠0 with probability (q−1)/q. Two families of state variables are introduced: the “drain” d(t) representing the deviation of linear complexity from its average, and the “batteries” b_m(t) representing deviations of auxiliary degrees. Their evolution obeys simple linear rules (equations (3) and (4) in the paper) and an invariant d + Σ_{m=1}^M b_m ≡ 0 (mod M+1) holds at every step. This invariant implies that the average of d and the b_m’s is zero, a fact that underpins the subsequent bounds.

Assuming that all M sequences are “active” (i.e., each experiences infinitely many non‑zero discrepancies), the authors prove the following tight inequalities:  M/(M+1) ≤ S ≤ 1,  M(1−S) ≤ I ≤ 1−S/M. Equivalently, for the normalized drain variables ˜I = I−M/(M+1) and ˜S = S−M/(M+1) one has  −M·˜S ≤ ˜I ≤ −˜S/M,  0 ≤ ˜S ≤ 1/(M+1). The proof proceeds in four parts: (a) the trivial bound S≤1 from L_a(n)≤n; (b) the lower bound S≥M/(M+1) by noting that the maximum of the batteries and the drain cannot stay below the average zero; (c) the lower bound on I using the fact that the sum of the normalized batteries is bounded by M·˜S; (d) the upper bound on I by analysing moments when the drain reaches its supremum ˜S and a battery discharges, leading to the inequality 0 ≥ ˜S+M·˜I. The authors also treat the case where only K≤M sequences are active, obtaining analogous bounds with M replaced by K.

To show that every admissible (I,S) pair is actually attainable, the paper presents an explicit construction based on a “hexagon” pattern. The time axis is divided into five intervals (t₀, t₁, tₓ, t₂, t*), during which the drain and batteries evolve with prescribed slopes (derived from equations (3)–(4)). In the first interval b₁ grows while the other batteries discharge; in the second all batteries grow; at tₓ the first battery reaches the target upper line ˜S·tₓ while the drain sits at ˜I·tₓ; thereafter the batteries are forced below the drain, preventing discharge; finally the remaining batteries discharge to bring all variables back to zero at t*. By carefully choosing the lengths of these intervals (expressed in terms of I, S, and M) the construction ensures that liminf and limsup of the normalized linear complexity converge exactly to the prescribed I and S. The process can be repeated indefinitely, yielding an infinite multisequence. When K<M, the authors simply set the extra M−K series to the zero series, which does not affect L_a(n).

The construction shows that for each admissible (I,S) there exist 2^{ℵ₀} distinct multisequences realizing it. Moreover, the set of such sequences has positive Hausdorff dimension whenever S<1, and its Haar measure is zero except for the unique point (I,S) = (M/(M+1), M/(M+1)), which corresponds to the typical behaviour of a random multisequence. This resolves an open problem posed by Dai, Imamura, and Yang, and extends earlier results that were limited to the single‑sequence case (M=1).

In the concluding discussion the authors note several implications. In cryptography, the linear complexity profile is a key indicator of unpredictability for stream ciphers; the results clarify the full range of possible asymptotic behaviours for parallel keystream generators. In coding theory and sequence design, the ability to engineer multisequences with prescribed liminf/limsup offers new flexibility for constructing sequences with tailored security or correlation properties. Finally, the battery‑discharge viewpoint provides a transparent probabilistic model that may be useful for analyzing other combinatorial or dynamical systems where a global invariant governs the evolution of multiple interacting quantities.