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The generalized method of solving ECDLP using quantum annealing

Łukasz Dzierzkowski

TL;DR

This paper presents a generalization of a method allowing the transformation of the Elliptic Curve Discrete Logarithm Problem over prime fields to the Quadratic Unconstrained Binary Optimization problem, which has no restriction that a given elliptic curve model has complete arithmetic.

Abstract

This paper presents a generalization of a method allowing the transformation of the Elliptic Curve Discrete Logarithm Problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. The original method requires that a given elliptic curve model has complete arithmetic. The new one has no such restriction, which is a breakthrough. Since the mentioned obstacle is no longer a problem, the latest version of the algorithm may be used for any elliptic curve model. As a result, one may use quantum annealing to solve ECDLP on any given model of elliptic curves.

The generalized method of solving ECDLP using quantum annealing

TL;DR

This paper presents a generalization of a method allowing the transformation of the Elliptic Curve Discrete Logarithm Problem over prime fields to the Quadratic Unconstrained Binary Optimization problem, which has no restriction that a given elliptic curve model has complete arithmetic.

Abstract

This paper presents a generalization of a method allowing the transformation of the Elliptic Curve Discrete Logarithm Problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. The original method requires that a given elliptic curve model has complete arithmetic. The new one has no such restriction, which is a breakthrough. Since the mentioned obstacle is no longer a problem, the latest version of the algorithm may be used for any elliptic curve model. As a result, one may use quantum annealing to solve ECDLP on any given model of elliptic curves.

Paper Structure

This paper contains 14 sections, 25 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. ECDLP
  4. QUBO
  5. Transformation
  6. Problems with the original method
  7. Generalization - introduction
  8. Generalization - first idea
  9. Generalization - second idea
  10. Practical example
  11. Short Weierstrass curve
  12. Explanation
  13. Experiment
  14. Conclusion and further work

Figures (4)

  • Figure 1: The decomposition of ECDLP wronski2024transformation.
  • Figure 2: The modified decomposition of ECDLP.
  • Figure 3: Connection between source variables.
  • Figure 4: Embedding of a problem equivalent to the problem of finding elliptic curve discrete logarithm over $\mathbb{F}_3$ on shorts Weierstrass curve to the D-Wave Advantage.