Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A provably quasi-polynomial algorithm for the discrete logarithm problem in finite fields of small characteristic

Guido Lido

TL;DR

This paper establishes a probabilistic quasi-polynomial-time algorithm for the discrete logarithm problem in finite fields of small characteristic by leveraging elliptic presentations of the field. The approach embeds the target field into a slightly larger field with a presentation based on an elliptic curve, and then performs a carefully controlled descent to relate logs of arbitrary elements to a small factor base of divisors on the curve. A two-stage descent combined with Galois-theoretic lemmas controls irreducibility and traps, enabling an index-calculus with a factor base of essential degrees up to $2^8$ and a zig-zag descent that yields the needed relations in time $q^{O(\,\log [K:\mathbb{F}_q] \,)}$. Consequently, for all finite fields of small characteristic, the discrete logarithm problem can be solved in quasi-polynomial time by embedding into an elliptic-presentation field and executing the descent algorithm. The results bridge prior heuristic progress with a rigorous complexity analysis and provide a blueprint for practical, albeit not optimized, implementations using elliptic presentations.

Abstract

We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially follow the heuristically quasi-polynomial algorithm presented by Barbulescu, Gaudry, Joux and Thome'. The main difference is to use a presentation of the finite field based on elliptic curves: the abundance of elliptic curves ensures the existence of such a presentation.

A provably quasi-polynomial algorithm for the discrete logarithm problem in finite fields of small characteristic

TL;DR

This paper establishes a probabilistic quasi-polynomial-time algorithm for the discrete logarithm problem in finite fields of small characteristic by leveraging elliptic presentations of the field. The approach embeds the target field into a slightly larger field with a presentation based on an elliptic curve, and then performs a carefully controlled descent to relate logs of arbitrary elements to a small factor base of divisors on the curve. A two-stage descent combined with Galois-theoretic lemmas controls irreducibility and traps, enabling an index-calculus with a factor base of essential degrees up to and a zig-zag descent that yields the needed relations in time . Consequently, for all finite fields of small characteristic, the discrete logarithm problem can be solved in quasi-polynomial time by embedding into an elliptic-presentation field and executing the descent algorithm. The results bridge prior heuristic progress with a rigorous complexity analysis and provide a blueprint for practical, albeit not optimized, implementations using elliptic presentations.

Abstract

We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially follow the heuristically quasi-polynomial algorithm presented by Barbulescu, Gaudry, Joux and Thome'. The main difference is to use a presentation of the finite field based on elliptic curves: the abundance of elliptic curves ensures the existence of such a presentation.
Paper Structure (20 sections, 11 theorems, 98 equations, 1 figure)

This paper contains 20 sections, 11 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Elliptic presentations
  4. Traps
  5. Divisors and discrete logarithm
  6. The main algorithm
  7. The descent procedure
  8. Intermezzo: a lemma in Galois theory
  9. Descent 3-to-2
  10. The definition of $\mathcal{C}$
  11. The irreducible components of $\mathcal{C}$
  12. $k$-rational points on $\mathcal{C}$ and where to find them
  13. Descent 4-to-3
  14. The definition of $\mathcal{C}$
  15. Irreducibility of a projection of $\mathcal{C}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

There exists a probabilistic algorithm, described in Section sec_algo, that solves the discrete logarithm problem in $K^\times$ for all finite fields $K$ of small characteristic in expected time

Figures (1)

  • Figure 1: The intersections $X_i$ of the curve $\mathcal{B}_P$ with certain lines $L_{i,j}, R_{i,j}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 2.2
  • Theorem 5.1
  • Corollary 5.2
  • Theorem 5.4
  • Proposition 6.2
  • Proposition 6.3
  • Proposition 7.1
  • Proposition 7.4
  • Lemma 7.7
  • ...and 5 more