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Fast Quantum Algorithm for Solving Multivariate Quadratic Equations

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

Published Aug 27, 2019• 2 citations
OpenAlex

Abstract

In August 2015 the cryptographic world was shaken by a sudden and surprising announcement by the US National Security Agency NSA concerning plans to transition to post-quantum algorithms. Since this announcement post-quantum cryptography has become a topic of primary interest for several standardization bodies. The transition from the currently deployed public-key algorithms to post-quantum algorithms has been found to be challenging in many aspects. In particular the problem of evaluating the quantum-bit security of such post-quantum cryptosystems remains vastly open. Of course this question is of primarily concern in the process of standardizing the post-quantum cryptosystems. In this paper we consider the quantum security of the problem of solving a system of {\it $m$ Boolean multivariate quadratic equations in $n$ variables} (\MQb); a central problem in post-quantum cryptography. When $n=m$, under a natural algebraic assumption, we present a Las-Vegas quantum algorithm solving \MQb{} that requires the evaluation of, on average, $O(2^{0.462n})$ quantum gates. To our knowledge this is the fastest algorithm for solving \MQb{}.

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In August 2015 the cryptographic world was shaken by a sudden and surprising announcement by the US National Security Agency NSA concerning plans to transition to post-quantum algorithms. Since this announcement post-quantum cryptography has become a topic of primary interest for several standardization bodies. The transition from the currently deployed public-key algorithms to post-quantum algorithms has been found to be challenging in many aspects. In particular the problem of evaluating the quantum-bit security of such post-quantum cryptosystems remains vastly open. Of course this question is of primarily concern in the process of standardizing the post-quantum cryptosystems. In this paper we consider the quantum security of the problem of solving a system of {\it mmm Boolean multivariate quadratic equations in nnn variables} (\MQb); a central problem in post-quantum cryptography. When n=mn=mn=m, under a natural algebraic assumption, we present a Las-Vegas quantum algorithm solving \MQb{} that requires the evaluation of, on average, O(20.462n)O(2^{0.462n})O(20.462n) quantum gates. To our knowledge this is the fastest algorithm for solving \MQb{}.
In August 2015 the cryptographic world was shaken by a sudden and surprising announcement by the US National Security Agency NSA concerning plans to transition to post-quantum algorithms. Since this announcement post-quantum cryptography has become a topic of primary interest for several standardization bodies. The transition from the currently deployed public-key algorithms to post-quantum algorithms has been found to be challenging in many aspects. In particular the problem of evaluating the quantum-bit security of such post-quantum cryptosystems remains vastly open. Of course this question is of primarily concern in the process of standardizing the post-quantum cryptosystems. In this paper we consider the quantum security of the problem of solving a system of {\it mmm Boolean multivariate quadratic equations in nnn variables} (\MQb); a central problem in post-quantum cryptography. When n=mn=mn=m, under a natural algebraic assumption, we present a Las-Vegas quantum algorithm solving \MQb{} that requires the evaluation of, on average, O(20.462n)O(2^{0.462n})O(20.462n) quantum gates. To our knowledge this is the fastest algorithm for solving \MQb{}.
The security of current public-key cryptosystems relies on the hardness of factoring large integers or solving discrete logarithm problems. However, these mathematical problems can be solved in polynomial time using a quantum computer. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. In this expository article, we give an overview of recent research on post-quantum cryptography. In particular, we describe the construction and security of multivariate polynomial cryptosystems and lattice-based cryptosystems, which are the main candidates of post-quantum cryptography.
The development of quantum computers has led to the possibility of solving problems intractable to deal with in clas-sical computers. That threatens the current standard of public-key cryptography based on factoring or discrete-log problem. As a result, it is necessary to use a secure cryptosystem even in a quantum computing environment. NIST has been conducting a Post-Quantum Cryptography (PQC) standardization process. Recently, there have been attempting to design binary MQ solver with the Grover's algorithm to solve the binary Multivariate Quadratic (MQ) problem. However, most PQC candidates based on the MQ problem do not select a small security parameter and are over GF(16) and above. In this paper, we design the Grover's oracle circuit to solve the MQ problem over GF(31) that can be used to analyze the quantum security of PQC candidates such as Rainbow and MQDSS.
This article analyzes the threats to the security of modern cryptographic systems posed by the development of quantum computers, as well as the emergence of the concept of post-quantum cryptography as a solution to these problems. The study demonstrates that classical public-key cryptosystems, particularly RSA, DSA, and elliptic curve-based algorithms, can be broken using quantum algorithms. At the same time, new cryptographic systems based on hash functions, codes, lattices, and multivariate quadratic equations are considered a promising direction. The main challenges related to the efficiency, reliability, and applicability of post-quantum cryptography are also highlighted. As a result, the necessity of developing new cryptographic standards to ensure future information security is demonstrated.
The advent of quantum computing presents a significant threat to classical public-key cryptographic systems, including RSA and Elliptic Curve Cryptography (ECC), thereby accelerating the development and standardization of post-quantum cryptography (PQC). This study evaluates the performance and implementation challenges of leading PQC algorithms across three major categories: lattice-based, hash-based, and multivariate cryptographic schemes. Utilizing empirical benchmark data from NIST candidate algorithms, the analysis focuses on key generation time, encryption latency, and memory consumption. Simulations and regression modeling reveal that lattice-based algorithms, particularly Kyber, demonstrate favorable trade-offs between computational efficiency and cryptographic strength. However, these schemes also present integration difficulties, especially in resource-constrained or legacy systems. Hash-based schemes like SPHINCS+ exhibit strong security guarantees but suffer from high latency and memory overhead, while multivariate algorithms such as Rainbow offer compact key sizes with moderate performance. The findings highlight the need for application-specific evaluation in selecting PQC solutions and provide actionable insights for guiding secure and efficient cryptographic transitions in the post-quantum era. Keywords: Post-Quantum Cryptography, Lattice-Based Cryptography, Hash-Based Cryptography, Multivariate Cryptography, Quantum Computing, Cryptographic Algorithms, Key Generation, Encryption Latency, Implementation Challenges, NIST PQC Standardization
The advent of quantum computing poses a significant threat to current cryptographic systems, particularly public-key algorithms such as RSA and ECC, which are vulnerable to quantum attacks like Shor's algorithm. Post-Quantum Cryptography (PQC) aims to develop and standardize quantum-resistant cryptographic algorithms to secure digital communications in the quantum era. This paper examines the current landscape of PQC, focusing on the evaluation and standardization efforts led by organizations such as the National Institute of Standards and Technology (NIST). We analyze the major families of quantum-resistant algorithms, including lattice-based, hash-based, code-based, multivariate, and isogeny-based cryptography, discussing their security guarantees, performance trade-offs, and implementation challenges. Furthermore, we review NIST's PQC standardization process, highlighting selected finalists and alternate candidates, and explore the ongoing transition challenges in adopting these algorithms across industries. Finally, we discuss future directions for PQC research, including hybrid cryptographic systems and the need for continued cryptanalysis to ensure long-term security. This work provides a comprehensive overview of the progress and challenges in deploying quantum-resistant cryptography to safeguard against emerging quantum threats.