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The Berlekamp-Massey Algorithm revisited

Nadia Ben Atti, Gema M. Diaz--Toca, Henri Lombardi

TL;DR

This work revisits the Berlekamp-Massey algorithm for extracting the minimal polynomial of a linearly recurrent sequence. It introduces a simple modification that eliminates the need to reverse the intermediate polynomial and connects the method to Hankel-matrix ranks, enabling a direct computation of the minimal polynomial and a natural lazy evaluation workflow. The paper also lays out a detailed lazy variant applicable to large-algebra contexts (à la Wiedemann) where only a partial coefficient window is processed initially, with reuse of prior Euclidean steps to progressively refine the result. Together, these contributions improve interpretability, potentially reduce coefficient requirements when the true degree is small, and provide a flexible framework for adaptive polynomial computation in algebraic settings.

Abstract

We propose a slight modification of the Berlekamp-Massey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation.

The Berlekamp-Massey Algorithm revisited

TL;DR

This work revisits the Berlekamp-Massey algorithm for extracting the minimal polynomial of a linearly recurrent sequence. It introduces a simple modification that eliminates the need to reverse the intermediate polynomial and connects the method to Hankel-matrix ranks, enabling a direct computation of the minimal polynomial and a natural lazy evaluation workflow. The paper also lays out a detailed lazy variant applicable to large-algebra contexts (à la Wiedemann) where only a partial coefficient window is processed initially, with reuse of prior Euclidean steps to progressively refine the result. Together, these contributions improve interpretability, potentially reduce coefficient requirements when the true degree is small, and provide a flexible framework for adaptive polynomial computation in algebraic settings.

Abstract

We propose a slight modification of the Berlekamp-Massey Algorithm for obtaining the minimal polynomial of a given linearly recurrent sequence. Such a modification enables to explain it in a simpler way and to adapt it to lazy evaluation.
Paper Structure (6 sections, 3 theorems, 19 equations, 3 algorithms)

This paper contains 6 sections, 3 theorems, 19 equations, 3 algorithms.

Table of Contents

  1. Introduction: The usual Berlekamp-Massey algorithm
  2. Some good reasons to modify the usual algorithm
  3. Lazy Evaluation
  4. Introduction: l'algo-rithme de Berlekamp-Massey usuel
  5. De bonnes raisons pour modifier l'algo-rithme usuel
  6. Une variante paresseuse

Key Result

Proposition 2.1

Let $\underline a$ be a linearly recurrent sequence. If $\underline a$ has a generating polynomial of degree $\leq n$, then the degree $d$ of its minimal polynomial ${\rm P}^{\underline a }$ is equal to the rank of the Hankel matrix The coefficients of ${\rm P}^{\underline a }(x)=x^d-\sum_{i=0}^{d-1}g_ix^i\in\mathbb{K}[x]\,$ are provided by the unique solution of the linear system that is,

Theorems & Definitions (4)

  • Proposition 2.1
  • Corollary 2.2
  • proof
  • Proposition 2.1