Extended Algorithm for Solving Underdefined Multivariate Quadratic Equations
It is well known that solving randomly chosen Multivariate Quadratic equations over a finite field (MQ-Problem) is NP-hard, and the security of Multivariate Public Key Cryptosystems (MPKCs) is based on the MQ-Problem. However, this problem can be solved efficiently when the number of unknowns n is sufficiently greater than that of equations m (This is called “Underdefined”). Indeed, the algorithm by Kipnis et al. (Eurocrypt’99) can solve the MQ-Problem over a finite field of even characteristic in a polynomial-time of n when n ≥ m(m + 1). Therefore, it is important to estimate the hardness of the MQ-Problem to evaluate the security of Multivariate Public Key Cryptosystems. We propose an algorithm in this paper that can solve the MQ-Problem in a polynomial-time of n when n ≥ m(m + 3)/2, which has a wider applicable range than that by Kipnis et al. We will also compare our proposed algorithm with other known algorithms. Moreover, we implemented this algorithm with Magma and solved the MQ-Problem of m = 28 and n = 504, and it takes 78.7 seconds on a common PC.
KeywordsMultivariate Public Key Cryptosystems (MPKCs) Multivariate Quadratic Equations MQ-Problem
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- 1.Bardet, M., Faugère, J.-C., Salvy, B., Yang, B.-Y.: Asymptotic Behaviour of the Degree of Regularity of Semi-Regular Polynomial Systems, MEGA 2005 (2005), http://www-polsys.lip6.fr/~jcf/Papers/BFS05b.pdf
- 4.Computational Algebra Group, University of Sydney. The MAGMA Computational Algebra System for Algebra, Number Theory, and GeometryGoogle Scholar
- 8.Ding, J., Gower, J.E., Schmidt, D.S.: Multivariate Public Key Cryptosystems. Springer (2006)Google Scholar
- 10.Faugère, J.-C.: “A New Efficient Algorithm for Computing Gröbner bases (F 4)”. Journal of Pure and Applied Algebra 139, 61–88 (1999)Google Scholar
- 11.Faugère, J.-C.: A New Efficient Algorithm for Computing Gröbner bases without reduction to zero (F 5). In: Proceedings of ISSAC 2002, pp. 75–83. ACM Press (2002)Google Scholar
- 12.Faugère, J.-C., Perret, L.: On the Security of UOV. In: Proceedings of SCC 2008, pp. 103–109 (2008)Google Scholar
- 13.Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H.Freeman (1979)Google Scholar
- 14.Hashimoto, Y.: Algorithms to Solve Massively Under-Defined Systems of Multivariate Quadratic Equations. IEICE Trans. Fundamentals E94-A(6), 1257–1262 (2011)Google Scholar
- 17.Patarin, J.: Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt ’88. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248–261. Springer, Heidelberg (1995)Google Scholar
- 19.Shor, P.W.: Algorithms for Quantum Computation: Discrete Logarithms and Factoring. In: Proceedings of 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. IEEE Computer Society Press (1994)Google Scholar