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 behavior of the normalized linear complexity of multisequences over a finite field F_q. A multisequence a = (a₁,…,a_M)∈F_q^∞ consists of M infinite component sequences. For each n, L_a(n) denotes the linear complexity, i.e., the length of the shortest linear feedback shift register (LFSR) that can generate the first n symbols of the whole multisequence. The authors focus on the two limits
 I = lim inf_{n→∞} L_a(n)/n and S = lim sup_{n→∞} L_a(n)/n,
which capture the lower and upper asymptotic density of the complexity.

A key hypothesis is that every component sequence produces a non‑zero discrepancy infinitely often; this guarantees that the complexity does not become stationary. Under this condition the authors introduce the Battery Discharge Model (BDM), a deterministic dynamical system that mimics the evolution of linear complexity. In the BDM each component sequence corresponds to a “battery” whose voltage changes according to a simple rule: when the discrepancy is zero the battery discharges (voltage decreases), otherwise it charges (voltage increases). An invariant of the model is the total voltage, which cannot exceed a constant C that depends only on M. By translating the invariant into inequalities for the normalized complexity, the authors obtain the fundamental bounds

 M/(M+1) ≤ S ≤ 1,

 M(1 − S) ≤ I ≤ 1 − S/M.

The lower bound for S reflects the worst‑case scenario where all batteries are almost simultaneously discharged, while the upper bound is trivial because the complexity can never exceed n. The relationship between I and S follows from the fact that the total voltage must stay within the invariant window; when S is close to 1 the lower limit I collapses to 0, and when S attains its minimal value M/(M+1) the upper limit for I approaches 1 − 1/(M+1).

Having identified the admissible region for (I,S), the authors turn to the constructive side: for any pair (I,S) satisfying the above inequalities they explicitly build multisequences that realize it. The construction uses the BDM’s state space, which can be encoded by an infinite binary selection sequence σ = (σ₁,σ₂,…). At each step σ_k = 0 or 1 determines whether a particular battery is forced to discharge or charge, thereby controlling the instantaneous slope of L_a(n). By carefully balancing the frequencies of 0’s and 1’s, one can force the empirical averages of L_a(n)/n to converge to any prescribed (I,S) within the admissible region. Because the binary selection sequence contains infinitely many free choices, there are continuum many (2^{ℵ₀}) distinct multisequences that share the same (I,S) pair. The mapping from σ to the resulting multisequence is an isometry, guaranteeing that distinct σ’s produce distinct complexity trajectories.

The paper also situates its results within the existing literature. For M = 1 the bounds reduce to the classical result ½ ≤ S ≤ 1 with I = 1 − S, confirming that the single‑sequence case is a special instance of the general theory. Moreover, the work resolves an open problem posed by Dai, Imamura, and Yang, who asked whether a complete description of the possible asymptotic linear‑complexity pairs for multisequences exists. The present paper answers affirmatively and provides a full characterization.

From a practical standpoint, the findings have several implications. In multistream cryptographic constructions, each stream must exhibit high linear complexity, yet the overall system should avoid pathological correlations; the derived bounds give precise quantitative limits for such designs. The constructive method shows how to generate families of multisequences with prescribed complexity profiles, which can be used to test randomness generators, to benchmark stream‑cipher algorithms, or to design sequences with tailored security properties. Finally, the BDM framework offers a new analytical tool that could be extended to other complexity measures (e.g., jump complexity or k‑error linear complexity) and to more general algebraic structures beyond finite fields.