On Sequences with a Perfect Linear Complexity Profile

TL;DR

This paper investigates sequences with a perfect linear complexity profile, providing new proofs, bounds, and characterizations, thereby advancing understanding of their algebraic structure and complexity behavior.

Contribution

It introduces new proofs for existing theorems, derives bounds on linear complexities, and offers novel characterizations of sequences with perfect linear complexity profiles.

Findings

Proves a theorem of Wang and Massey using Bézout identities.

Provides a tight upper bound for the sum of linear complexities.

Characterizes sequences with perfect linear complexity profiles in new ways.

Abstract

We derive B\'ezout identities for the minimal polynomials of a finite sequence and use them to prove a theorem of Wang and Massey on binary sequences with a perfect linear complexity profile. We give a new proof of Rueppel's conjecture and simplify Dai's original proof. We obtain short proofs of results of Niederreiter relating the linear complexity of a sequence s and K(s), which was defined using continued fractions. We give an upper bound for the sum of the linear complexities of any sequence. This bound is tight for sequences with a perfect linear complexity profile and we apply it to characterise these sequences in two new ways.