An Algorithm for Computing $m$-Tight Error Linear Complexity of Sequences over $GF(p^{m})$ with Period $p^{m}$

The linear complexity (LC) of a sequence has been used as a convenient measure of the randomness of a sequence. Based on the theories of linear complexity, $k$-error linear complexity, the minimum error and the $k$-error linear complexity profile, the notion of $m$-tight error linear complexity is presented. An efficient algorithm for computing $m$-tight error linear complexity is derived from the algorithm for computing $k$-error linear complexity of sequences over GF($p^{m}$) with period $p^n$, where $p$ is a prime. The validity of the algorithm is shown. The algorithm is also realized with C language, and an example is presented to illustrate the algorithm.

💡 Research Summary

The paper addresses the problem of assessing the stability of linear complexity (LC) for periodic sequences over the finite field GF(p^m) with period N = p^n, where p is a prime. While a large LC is a traditional indicator of randomness and cryptographic strength, it does not guarantee security because a small number of symbol changes can cause a dramatic drop in LC. To quantify this “stability” the authors build on several existing notions: k‑error linear complexity (the smallest LC achievable after changing at most k symbols), the minimum error minerror(S) (the smallest k that forces a strict decrease of LC), and the k‑error LC profile (the sequence of LC values as k increases).

The state‑of‑the‑art method for computing k‑error LC is the Stamp‑Martin algorithm, which is a binary‑specific adaptation of the Games‑Chan algorithm. For sequences over GF(p^m) with period p^n, Kaida, Uehara and Imamura (KUI) generalized this approach, achieving an O(p·log_p N) time algorithm that works by recursively splitting the period‑p^n sequence into p blocks, applying a set of transformation functions F_u, and maintaining two auxiliary matrices: AC(M) (the minimum number of changes needed to alter a particular element of the original sequence) and BC(M) (the minimum number of changes needed to force a set of transformed components to zero). The KUI algorithm can compute the k‑error LC for any given k, but to obtain the whole profile one would have to invoke it repeatedly for many k values, which becomes costly.

The main contribution of the present work is the introduction of m‑tight error linear complexity (m‑tight error LC), denoted as a pair (k_m, C_m). Here k_m is the smallest number of symbol modifications required to cause the m‑th “jump” (i.e., the m‑th decrease) in the LC profile, and C_m is the resulting LC after that jump. Thus, the m‑tight error LC captures the m‑th point of the k‑error LC profile in a single compact representation.

To compute this new metric efficiently, the authors modify the KUI algorithm (Algorithm 2.2) by adding two simple bookkeeping steps:

1. T_min – a variable that records, during each recursion level, the smallest number of changes that would force a decrease of LC at the current depth. It is updated whenever a candidate value TB