Abstract
MutantXL is an algorithm for solving systems of polynomial equations that was proposed at SCC 2008. This paper proposes two substantial improvements to this algorithm over GF(2) that result in significantly reduced memory usage. We present experimental results comparing MXL2 to the XL algorithm, the MutantXL algorithm and Magma’s implementation of F 4. For this comparison we have chosen small, randomly generated instances of the MQ problem and quadratic systems derived from HFE instances. In both cases, the largest matrices produced by MXL2 are substantially smaller than the ones produced by MutantXL and XL. Moreover, for a significant number of cases we even see a reduction of the size of the largest matrix when we compare MXL2 against Magma’s F 4 implementation.
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References
Matsumoto, T., Imai, H.: Public Quadratic Polynomial-Tuples for Efficient Signature-Verification and Message-Encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419–453. Springer, Heidelberg (1988)
Patarin, J.: Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of Asymmetric Algorithms. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 33–48. Springer, Heidelberg (1996)
Patarin, J., Goubin, L., Courtois, N.: \(C^{*}_{-+}\) and HM: Variations Around Two Schemes of T. Matsumoto and H. Imai. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 35–50. Springer, Heidelberg (1998)
Moh, T.: A Public Key System With Signature And Master Key Functions. Communications in Algebra 27, 2207–2222 (1999)
Ding, J.: A New Variant of the Matsumoto-Imai Cryptosystem through Perturbation. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 305–318. Springer, Heidelberg (2004)
Courtois, N.T., Klimov, A., Patarin, J., Shamir, A.: Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 392–407. Springer, Heidelberg (2000)
Ding, J., Buchmann, J., Mohamed, M.S.E., Moahmed, W.S.A., Weinmann, R.P.: MutantXL. In: Proceedings of the 1st international conference on Symbolic Computation and Cryptography (SCC 2008), Beijing, China, LMIB, pp. 16–22 (2008), http://www.cdc.informatik.tu-darmstadt.de/reports/reports/MutantXL_Algorithm.pdf
Ding, J., Cabarcas, D., Schmidt, D., Buchmann, J., Tohaneanu, S.: Mutant Gröbner Basis Algorithm. In: Proceedings of the 1st international conference on Symbolic Computation and Cryptography (SCC 2008), Beijing, China, LMIB, pp. 23–32 (2008)
Courtois, N.T.: Experimental Algebraic Cryptanalysis of Block Ciphers (2007), http://www.cryptosystem.net/aes/toyciphers.html
Segers, A.: Algebraic Attacks from a Gröbner Basis Perspective. Master’s thesis, Department of Mathematics and Computing Science, TECHNISCHE UNIVERSITEIT EINDHOVEN, Eindhoven (2004)
Shigeo, M.: Hotaru (2005), http://cvs.sourceforge.jp/cgi-bin/viewcvs.cgi/hotaru/hotaru/hfe25-96?view=markup
Albrecht, M., Bard, G.: M4RI – Linear Algebra over GF(2) (2008), http://m4ri.sagemath.org/index.html
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Mohamed, M.S.E., Mohamed, W.S.A.E., Ding, J., Buchmann, J. (2008). MXL2: Solving Polynomial Equations over GF(2) Using an Improved Mutant Strategy. In: Buchmann, J., Ding, J. (eds) Post-Quantum Cryptography. PQCrypto 2008. Lecture Notes in Computer Science, vol 5299. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88403-3_14
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DOI: https://doi.org/10.1007/978-3-540-88403-3_14
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