A New Hybrid Algorithm for Multivariate Polynomial System Solving
Abstract
The method of solving a polynomial system of equations plays a crucial role in various domains of cryptography. For instance, multivariate cryptography using public keys relies on the computational difficulty of solving a multivariate polynomial problem over a finite field. Aram Harrow, Avinatan Hassidam, and Seth Loyd proposed a quantum algorithm (HHL) to solve the Equation Ax = b, where A is a Hermitian matrix. Compared to the fastest classical approach, which has an run time, the HHL algorithm has a poly runtime that offers exponential speed-up. To solve a set of multivariate polynomial equations, we turned to the HHL method for assistance. In particular, our proposed algorithm will be a building block for future quantum XL algorithms. Here, we have proposed a new hybrid quantum XL algorithm and used it to solve a standard problem to demonstrate its efficacy.
References
[1]
Adams WW, Loustaunau P. An introduction to Gröbner bases, vol. 3. American Mathematical Society; 2022.
[2]
Ambainis A. Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations. 2010. arXiv:1010.4458.
[3]
Bard GV, Courtois NT, Jefferson C. Efficient methods for conversion and solution of sparse systems of low-degree multivariate polynomials over gf (2) via sat-solvers. Cryptology ePrint Archive. 2007.
[4]
Bard G. Algebraic cryptanalysis. Springer, New York; 2009.
[5]
Bouillaguet C, Chen HC, Cheng CM, Chou T, Niederhagen R, Shamir A, Yang BY. Fast exhaustive search for polynomial systems in. In: International workshop on cryptographic hardware and embedded systems. Springer. 2010. pp. 203–18.
[6]
Buchberger B and Winkler F Gröbner bases and applications 1998 Cambridge Cambridge University Press
[7]
Chen YA, Gao XS. Quantum algorithms for boolean equation solving and quantum algebraic attack on cryptosystems. 2018.
[8]
Childs AM, Kothari R. Limitations on the simulation of non-sparse Hamiltonians. 2009. arXiv:0908.4398.
[9]
Childs AM, Kothari R, and Somma RD Quantum algorithm for systems of linear equations with exponentially improved dependence on precision SIAM J Comput 2017 46 6 1920-1950
[10]
Courtois N. The security of cryptographic primitives based on multivariate algebraic problems: Mq, Minrank, Ip, Hfe. Paris: Paris 6 University; 2001.
[11]
Courtois N, Klimov A, Patarin J, Shamir A. Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Preneel B, editor. Advances in cryptology—EUROCRYPT 2000. Berlin: Springer; 2000. pp. 392–407.
[12]
Courtois NT, Patarin J. About the xl algorithm over gf (2). In: Topics in cryptology-CT-RSA 2003: the cryptographers’ track at the RSA conference 2003 San Francisco, CA, USA, April 13–17, 2003 Proceedings. Springer; 2003. pp. 141–57.
[13]
Diem C. The xl-algorithm and a conjecture from commutative algebra. In: Advances in cryptology-ASIACRYPT 2004: 10th international conference on the theory and application of cryptology and information security, Jeju Island, Korea, December 5–9, 2004. Proceedings 10. Springer; 2004. pp. 323–37.
[14]
Ding J, Gheorghiu V, Gilyén A, Hallgren S, and Li J Limitations of the Macaulay matrix approach for using the hhl algorithm to solve multivariate polynomial systems Quantum 2023 7 1069
[15]
Grover LK. A fast quantum mechanical algorithm for database search. 1996.
[16]
Harrow AW, Hassidim A, and Lloyd S Quantum algorithm for linear systems of equations Phys Rev Lett 2009 103 15
[17]
Lee Y, Joo J, and Lee S Hybrid quantum linear equation algorithm and its experimental test on ibm quantum experience Sci Rep 2019 9 1 4778
[18]
Livingston WP, Blok MS, Flurin E, Dressel J, Jordan AN, and Siddiqi I Experimental demonstration of continuous quantum error correction Nat Commun 2022 13 1 2307
[19]
Moh T On the method of “xl” and its inefficiency to ttm IACR Cryptol ePrint Arch 2001 2001 47
[20]
Preskill J Quantum computing in the nisq era and beyond Quantum 2018 2 79
[21]
Shor PW Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer SIAM J Comput 1997 26 5 1484-1509
[22]
Xiao L Applicability of xsl attacks to block ciphers Electron Lett 2003 39 25 1
[23]
Zhang M, Dong L, Zeng Y, Cao N. Circuit implementation of the globally optimized hhl algorithm and experiments on qiskit. Researchsquare. 2022.
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Publication History
Published: 27 March 2024
Accepted: 20 January 2024
Received: 19 December 2023
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