Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Electronics and Telecommunications

Transformation of the discrete logarithm problem over F2n to the QUBO problem using normal bases

Michał Wroński, Mateusz Leśniak

Published Apr 7, 2025• 0 citations
Publisher

Abstract

Quantum computations are very important branch of modern cryptology. According to the number of working physical qubits available in general-purpose quantum computers and in quantum annealers, there is no coincidence, that nowadays quantum annealers allow to solve larger problems. In this paper we focus on solving discrete logarithm problem (DLP) over binary fields using quantum annealing. It is worth to note, that however solving DLP over prime fields using quantum annealing has been considered before, no author, until now, has considered DLP over binary fields using quantum annealing. Therefore, in this paper, we aim to bridge this gap. We present a polynomial transformation of the discrete logarithm problem over binary fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem, using approximately 3n2 logical variables for the binary field F2n. In our estimations, we assume the existence of an optimal normal base in the given fields. Such a QUBO instance can then be solved using quantum annealing.

DOI
10.24425/ijet.2025.153543

Related papers

Transformation of the discrete logarithm problem over $\mathbb F_{2^n}$ to the QUBO problem using normal bases

Michał Wroński, Mateusz Leśniak

Quantum computations are very important branch of modern cryptology. According to the number of working physical qubits available in general-purpose quantum computers and in quantum annealers, there is no coincidence, that nowadays quantum annealers allow to solve larger problems. In this paper we focus on solving discrete logarithm problem (DLP) over binary fields using quantum annealing. It is worth to note, that however solving DLP over prime fields using quantum annealing has been considered before, no author, until now, has considered DLP over binary fields using quantum annealing. Therefore, in this paper, we aim to bridge this gap. We present a polynomial transformation of the discrete logarithm problem over binary fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem, using approximately $3n^2$ logical variables for the binary field $\mathbb{F}_{2^n}$. In our estimations, we assume the existence of an optimal normal base of II type in the given fields. Such a QUBO instance can then be solved using quantum annealing.

arXiv:2409.18534•Sep 27, 2024•0 citations

Transformation of Elliptic Curve Discrete Logarithm Problem to QUBO Using Direct Method in Quantum Annealing Applications

Michał Wroński, Elżbieta Burek, Łukasz Dzierzkowski +1 more

This paper investigates how to reduce the elliptic curve discrete logarithm problem over prime fields to the quadratic unconstrained binary optimization (QUBO) problem in order to obtain as few logical qubits as possible. In the best case scenario, if n is the bitlength of a characteristic of prime field Fp, approximately 3n³ logical qubits are required for such a reduction in the Edwards curve case. We present a practical attack on an elliptic curve discrete logarithm problem over the 3-bit prime field F7 for an elliptic curve with the subgroup of order 8. We solved this problem using the D-Wave Advantage QPU. To the best of the authors' knowledge, no one has made, so far, a practical attack on the elliptic curve discrete logarithm over a prime field using the direct quantum method.

Journal of Telecommunications and Information Technology•Feb 21, 2024•3 citations

Base of exponent representation matters -- more efficient reduction of discrete logarithm problem and elliptic curve discrete logarithm problem to the QUBO problem

Michał Wroński, Łukasz Dzierzkowski

This paper presents further improvements in the transformation of the Discrete Logarithm Problem (DLP) and Elliptic Curve Discrete Logarithm Problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. This is significant from a cryptanalysis standpoint, as QUBO problems may be solved using quantum annealers, and the fewer variables the resulting QUBO problem has, the less time is expected to obtain a solution.The main idea presented in the paper is allowing the representation of the exponent in different bases than the typically used base 2 (binary representation). It is shown that in such cases, the reduction of the discrete logarithm problem over the prime field \(\mathbb{F}_p\) to the QUBO problem may be obtained using approximately \(1.89n^2\) logical variables for \(n\) being the bitlength of prime \(p\), instead of the \(2n^2\) which was previously the best-known reduction method. The paper provides a practical example using the given method to solve the discrete logarithm problem over the prime field \(\mathbb{F}_{47}\). Similarly, for the elliptic curve discrete logarithm problem over the prime field \(\mathbb{F}_p\), allowing the representation of the exponent in different bases than typically used base two results in a lower number of required logical variables for \(n\) being the bitlength of prime \(p\), from \(3n^3\) to \(\frac{6n^3}{\log_{2}\left(\frac{3}{4}n\right)}\) logical variables, in the case of Edwards curves.

Quantum Information and Computation•Jun 1, 2024•2 citations

The generalized method of solving ECDLP using quantum annealing

Łukasz Dzierzkowski

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.

International Journal of Electronics and Telecommunications•Apr 7, 2025•1 citation

The generalized method of solving ECDLP using quantum annealing

Łukasz Dzierzkowski

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.

arXiv:2410.08725•Oct 11, 2024•1 citation

New QUBO Transformations to Improve Quantum and Simulated Annealing Performance for Quadratic Knapsack

Nicolás Borrajo, Juan Ramírez, Farzam Nosrati +3 more

Recent advancements in quantum computing have demonstrated significant potential for solving combinatorial optimization problems, like the quadratic knapsack problem, a constrained binary optimization problem. However, current quantum and quantum-inspired algorithms often require transforming these constrained problems into an unconstrained form, known as Quadratic Unconstrained Binary Optimization (QUBO). Such transformations can significantly impact the algorithms’ speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess all nine methods using Simulated Annealing and find that three of our approaches outperform existing methods in terms of execution time and the quality and quantity of feasible solutions found. Additionally, we tested these transformations on quantum annealers, which were unable to solve even small problem instances, due to limitations in connectivity and error rates. However, our results highlight the advantages of the new approaches, which reduce the total number of variables in the QUBO representation. This is a critical factor for enhanced performance on emerging quantum hardware, since it also reduces the required number of qubits and the embedding chain lengths.

International Conference on Agents and Artificial Intelligence (ICAART)•Jan 1, 2026•1 citation
Quantum computations are very important branch of modern cryptology. According to the number of working physical qubits available in general-purpose quantum computers and in quantum annealers, there is no coincidence, that nowadays quantum annealers allow to solve larger problems. In this paper we focus on solving discrete logarithm problem (DLP) over binary fields using quantum annealing. It is worth to note, that however solving DLP over prime fields using quantum annealing has been considered before, no author, until now, has considered DLP over binary fields using quantum annealing. Therefore, in this paper, we aim to bridge this gap. We present a polynomial transformation of the discrete logarithm problem over binary fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem, using approximately 3n2 logical variables for the binary field F2n. In our estimations, we assume the existence of an optimal normal base in the given fields. Such a QUBO instance can then be solved using quantum annealing.
Transformation of the discrete logarithm problem over F2n\mathbb F_{2^n}F2n​ to the QUBO problem using normal bases
Quantum computations are very important branch of modern cryptology. According to the number of working physical qubits available in general-purpose quantum computers and in quantum annealers, there is no coincidence, that nowadays quantum annealers allow to solve larger problems. In this paper we focus on solving discrete logarithm problem (DLP) over binary fields using quantum annealing. It is worth to note, that however solving DLP over prime fields using quantum annealing has been considered before, no author, until now, has considered DLP over binary fields using quantum annealing. Therefore, in this paper, we aim to bridge this gap. We present a polynomial transformation of the discrete logarithm problem over binary fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem, using approximately 3n23n^23n2 logical variables for the binary field F2n\mathbb{F}_{2^n}F2n​. In our estimations, we assume the existence of an optimal normal base of II type in the given fields. Such a QUBO instance can then be solved using quantum annealing.
This paper investigates how to reduce the elliptic curve discrete logarithm problem over prime fields to the quadratic unconstrained binary optimization (QUBO) problem in order to obtain as few logical qubits as possible. In the best case scenario, if n is the bitlength of a characteristic of prime field Fp, approximately 3n³ logical qubits are required for such a reduction in the Edwards curve case. We present a practical attack on an elliptic curve discrete logarithm problem over the 3-bit prime field F7 for an elliptic curve with the subgroup of order 8. We solved this problem using the D-Wave Advantage QPU. To the best of the authors' knowledge, no one has made, so far, a practical attack on the elliptic curve discrete logarithm over a prime field using the direct quantum method.
This paper presents further improvements in the transformation of the Discrete Logarithm Problem (DLP) and Elliptic Curve Discrete Logarithm Problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. This is significant from a cryptanalysis standpoint, as QUBO problems may be solved using quantum annealers, and the fewer variables the resulting QUBO problem has, the less time is expected to obtain a solution.The main idea presented in the paper is allowing the representation of the exponent in different bases than the typically used base 2 (binary representation). It is shown that in such cases, the reduction of the discrete logarithm problem over the prime field Fp\mathbb{F}_pFp​ to the QUBO problem may be obtained using approximately 1.89n21.89n^21.89n2 logical variables for nnn being the bitlength of prime ppp, instead of the 2n22n^22n2 which was previously the best-known reduction method. The paper provides a practical example using the given method to solve the discrete logarithm problem over the prime field F47\mathbb{F}_{47}F47​. Similarly, for the elliptic curve discrete logarithm problem over the prime field Fp\mathbb{F}_pFp​, allowing the representation of the exponent in different bases than typically used base two results in a lower number of required logical variables for nnn being the bitlength of prime ppp, from 3n33n^33n3 to 6n3log⁡2(34n)\frac{6n^3}{\log_{2}\left(\frac{3}{4}n\right)}log2​(43​n)6n3​ logical variables, in the case of Edwards curves.
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.
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.
Recent advancements in quantum computing have demonstrated significant potential for solving combinatorial optimization problems, like the quadratic knapsack problem, a constrained binary optimization problem. However, current quantum and quantum-inspired algorithms often require transforming these constrained problems into an unconstrained form, known as Quadratic Unconstrained Binary Optimization (QUBO). Such transformations can significantly impact the algorithms’ speed and efficiency. In this study, we evaluate five existing transformation methods and propose four novel approaches. We assess all nine methods using Simulated Annealing and find that three of our approaches outperform existing methods in terms of execution time and the quality and quantity of feasible solutions found. Additionally, we tested these transformations on quantum annealers, which were unable to solve even small problem instances, due to limitations in connectivity and error rates. However, our results highlight the advantages of the new approaches, which reduce the total number of variables in the QUBO representation. This is a critical factor for enhanced performance on emerging quantum hardware, since it also reduces the required number of qubits and the embedding chain lengths.