The $k$-error linear complexity distribution for $2^n$-periodic binary sequences

The linear complexity and the $k$-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By studying the linear complexity of binary sequences with period $2^n$, one could convert the computation of $k$-error linear complexity into finding error sequences with minimal Hamming weight. Based on Games-Chan algorithm, the $k$-error linear complexity distribution of $2^n$-periodic binary sequences is investigated in this paper. First, for $k=2,3$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic balanced binary sequences (with linear complexity less than $2^n$) are characterized. Second, for $k=3,4$, the complete counting functions on the $k$-error linear complexity of $2^n$-periodic binary sequences with linear complexity $2^n$ are presented. Third, as a consequence of these results, the counting functions for the number of $2^n$-periodic binary sequences with the $k$-error linear complexity for $k = 2$ and 3 are obtained. Further more, an important result in a recent paper is proved to be not completely correct.

💡 Research Summary

The paper investigates the distribution of k‑error linear complexity (Lₖ) for binary sequences with period 2ⁿ, a topic of central importance in the security analysis of stream ciphers based on linear feedback shift registers (LFSRs). While the ordinary linear complexity L(s) measures the length of the shortest LFSR that generates a sequence s, the k‑error linear complexity Lₖ(s) is defined as the smallest linear complexity achievable after changing at most k bits within one period. High L(s) alone does not guarantee resistance; a sequence whose Lₖ drops dramatically for small k is cryptographically weak.

The authors adopt a novel two‑step approach. First, they use the Games‑Chan algorithm to compute L(s) efficiently for 2ⁿ‑periodic binary sequences. This algorithm reveals that the computation of Lₖ can be reduced to finding error sequences u with minimal Hamming weight whose addition to s yields a new sequence with lower linear complexity. Second, they observe a parity‑based simplification: when k is even, Lₖ(s) equals the (k‑1)‑error linear complexity, and when k is odd, Lₖ(s) equals the (k‑1)‑error linear complexity of the sequence obtained after one error. This decomposition allows the authors to treat the problem as two independent sub‑problems with reduced combinatorial complexity.

The paper is organized as follows. Section II collects necessary preliminaries: definitions of generating functions, minimal polynomials, and three key lemmas concerning the relationship between Hamming weight, linear complexity, and the effect of adding elementary error sequences Eᵢ (a single 1 at position i). Lemma 2.1 states that a 2ⁿ‑periodic binary sequence has maximal linear complexity 2ⁿ if and only if its period contains an odd number of ones. Lemma 2.2 describes how the linear complexity of the sum of two sequences behaves depending on whether their complexities are equal. Lemma 2.3 and Lemma 2.4 give explicit formulas for the linear complexity of sequences with two and four non‑zero positions, respectively.

In Section III the authors focus on sequences whose linear complexity is strictly less than 2ⁿ (balanced sequences). They treat k = 2 and k = 3. Using Lemma 3.3 they show that for sequences with L(s) = 2ⁿ − 1 − 2ᵐ (0 ≤ m < n − 1) the 2‑error linear complexity can be reduced only when an error sequence of weight two aligns with a specific distance pattern (distance 2ⁿ − m·(2a + 1) or 2ⁿ − 1). Lemma 3.5 then counts all such error patterns, leading to the closed‑form expression

N₂(2ⁿ − 1 − 2ᵐ) = (1 + C(2ⁿ,2) − 3·2^{n+m‑3})·2^{2ⁿ‑1‑2ⁿ‑m‑1}.

For k = 3 they exploit the parity observation: the 3‑error linear complexity of a sequence with L(s) < 2ⁿ coincides with its 2‑error linear complexity, so the same counting formulas apply.

Section IV treats the complementary case where the original sequence has maximal linear complexity L(s) = 2ⁿ. Here any single error flips the Hamming weight to even, immediately reducing the linear complexity to 2ⁿ − 1. The authors prove that for odd k (k = 3) the k‑error linear complexity equals the (k − 1)‑error linear complexity, and for even k (k = 4) it equals the (k − 1)‑error linear complexity of the sequence after one error. By constructing appropriate weight‑two error sequences and applying Lemmas 2.3–2.4, they derive explicit counting formulas for the number of sequences whose 3‑error (and 4‑error) linear complexities assume each possible value.

Section V combines the results of Sections III and IV to obtain the complete distribution of L₂ and L₃ for all 2ⁿ‑periodic binary sequences. In doing so, the authors identify a flaw in the earlier work of Kavuluru (references