Fast Quantum Algorithm for Solving Multivariate Quadratic Equations

Faugère, Jean-Charles; Horan, Kelsey; Kahrobaei, Delaram; Kaplan, Marc; Kashefi, Elham; Perret, Ludovic

doi:10.48550/arxiv.1712.07211

Cited by 5 publications

(7 citation statements)

References 22 publications

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“…Another approach which is worth exploring is Zhuang-Zi [16] which provides a new technique to solve system of multivariate polynomial equations over a finite field. In the post quantum paradigm, efficient methods for solving these system of equations are illustrated in [11] (only for boolean case), and [13] (linear system of equations). [12] entails a quantum algorithm for solving non linear system of equations in GF(q = p k ) which might be used for improving the proposed method in the future.…”

Section: Discussionmentioning

confidence: 99%

“…We have implemented the intersection method with the help of MAGMA [10]. In the post quantum era, we expect that we can solve the system of equations using the methods as illustrated in [11] [12] [13]. We do not expect that the intersection method would be an alternative to Shor's algorithm for solving EC-DLP [14] in the postquantum paradigm.…”

Section: Introductionmentioning

confidence: 99%

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A New Method for Geometric Interpretation of Elliptic Curve Discrete Logarithm Problem

Di Tullio,

Pal

2019

Preprint

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In this paper, we intend to study the geometric meaning of the discrete logarithm problem defined over an Elliptic Curve. The key idea is to reduce the Elliptic Curve Discrete Logarithm Problem (EC-DLP) into a system of equations. These equations arise from the interesection of quadric hypersurfaces in an affine space of lower dimension. In cryptography, this interpretation can be used to design attacks on EC-DLP. Presently, the best known attack algorithm having a sub-exponential time complexity is through the implementation of Summation Polynomials and Weil Descent. It is expected that the proposed geometric interpretation can result in faster reduction of the problem into a system of equations. These overdetermined system of equations are hard to solve. We have used F4 (Faugere) algorithms and got results for primes less than 500,000. Quantum Algorithms can expedite the process of solving these over-determined system of equations. In the absence of fast algorithms for computing summation polynomials, we expect that this could be an alternative. We do not claim that the proposed algorithm would be faster than Shor's algorithm for breaking EC-DLP but this interpretation could be a candidate as an alternative to the 'summation polynomial attack' in the post-quantum era.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A New Method for Geometric Interpretation of Elliptic Curve Discrete Logarithm Problem

Di Tullio,

Pal

2019

Preprint

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“…In Ref. [22], the authors present a quantum version of BooleanSolve [23], which is currently the fastest asymptotic algorithm for classically solving systems of non-linear Boolean equations, that takes advantage of Grover's quantum algorithm. Note that Ref.…”

Section: Introductionmentioning

confidence: 99%

“…[24] also proposed a new Gröbner-based quantum algorithm for solving quadratic equations with a complexity comparable to QuantumBooleanSolve (we refer to Ref. [22] for further details). Finally, in Ref.…”

Section: Introductionmentioning

confidence: 99%

Solving systems of Boolean multivariate equations with quantum annealing

Ramos-Calderer,

Bravo-Prieto,

Lin

et al. 2021

Preprint

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Polynomial systems over the binary field have important applications, especially in symmetric and asymmetric cryptanalysis, multivariate-based post-quantum cryptography, coding theory, and computer algebra. In this work, we study the quantum annealing model for solving Boolean systems of multivariate equations of degree 2, usually referred to as the Multivariate Quadratic problem. We present different methodologies to embed the problem into a Hamiltonian that can be solved by available quantum annealing platforms. In particular, we provide three embedding options, and we highlight their differences in terms of quantum resources. Moreover, we design a machineagnostic algorithm that adopts an iterative approach to better solve the problem Hamiltonian by repeatedly reducing the search space. Finally, we use D-Wave devices to successfully implement our methodologies on several instances of the Multivariate Quadratic problem.

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“…So any problem relates to searching can be improved directly by Grover's algorithm in quantum computer. More importantly, Grover's algorithm can achieve speedup (sometimes not just quadratic speedup) to many NP complete problems, such as 3-SAT [2], existence of Hamiltonian cycle [29], quadratic Boolean equations solving [16], etc. Unfortunately, it has been shown that Grover's algorithm is optimal [5,7], so there does not exist more efficient quantum algorithms to solve the searching problem.…”

Section: Introductionmentioning

confidence: 99%

From linear combination of quantum states to Grover's searching algorithm

Shao¹

2018

Preprint

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Linear combination of unitaries (LCU for short) is one of the most important techniques in designing quantum algorithms. In this paper, we propose a new quantum algorithm in three different forms to achieve LCU. Different from previous algorithms [14,15,27], the complexity now only depends on the number of the unitaries and the precision. So it will play more important role in the design of quantum algorithms when the number of unitaries is small, such as quantum iteration algorithms. Moreover, as an application of the new LCU, three new quantum algorithms to the searching problem will be proposed, which will provide us new insights into Grover's searching algorithm. We also show that the problem of LCU is closely related to the problem of if we can efficiently implement U t for 0 < t < 1 when U is an efficiently implemented unitary operator? This problem is not hard to solve. However, it becomes inefficient when it contains a strict requirement on precision, such as in Grover's algorithm. Finally, as an application of the new LCU technique, we will show that the quantum state of any real classical vector can be prepared efficiently in quantum computer. So this solves the "input problem" in quantum computer efficiently.

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Fast Quantum Algorithm for Solving Multivariate Quadratic Equations

Cited by 5 publications

References 22 publications

A New Method for Geometric Interpretation of Elliptic Curve Discrete Logarithm Problem

A New Method for Geometric Interpretation of Elliptic Curve Discrete Logarithm Problem

Solving systems of Boolean multivariate equations with quantum annealing

From linear combination of quantum states to Grover's searching algorithm

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