Abstract
We propose the idea of building a secure hash using quadratic or higher degree multivariate polynomials over a finite field as the compression function. We analyze some security properties and potential feasibility, where the compression functions are randomly chosen high-degree polynomials, and show that under some plausible assumptions, high-degree polynomials as compression functions has good properties. Next, we propose to improve on the efficiency of the system by using some specially designed polynomials generated by a small number of random parameters, where the security of the system would then relies on stronger assumptions, and we give empirical evidence for the validity of using such polynomials.
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References
Aumasson, J.-P., Meier, W.: Analysis of multivariate hash functions. In: Proc. ICISC. LNCS (to appear, 2007), cf. http://www.131002.net/files/pub/AM07.pdf
Bardet, M., Faugère, J.-C., Salvy, B.: On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In: Proceedings of the International Conference on Polynomial System Solving, pp. 71–74 (2004); Previously INRIA report RR-5049
Berbain, C., Gilbert, H., Patarin, J.: QUAD: A Practical Stream Cipher with Provable Security. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 109–128. Springer, Heidelberg (2006)
Computational Algebra Group, University of Sydney. The MAGMA Computational Algebra System for Algebra, Number Theory and Geometry http://magma.maths.usyd.edu.au/magma/
Courtois, N., Goubin, L., Meier, W., Tacier, J.-D.: Solving Underdefined Systems of Multivariate Quadratic Equations. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, Springer, Heidelberg (2002)
Ding, J., Gower, J., Schmidt, D.: Multivariate Public-Key Cryptosystems. In: Advances in Information Security, Springer, Heidelberg (2006)
Garey, M.R., Johnson, D.S.: Computers and Intractability — A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)
Kipnis, A., Shamir, A.: Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. In: Wiener, M.J. (ed.) CRYPTO 1999. LNCS, vol. 1666, Springer, Heidelberg (1999), http://www.minrank.org/hfesubreg.ps or http://citeseer.nj.nec.com/kipnis99cryptanalysis.html
Menezes, A., Koblitz, N.: Another Look at “Provable Security”. II. In: Barua, R., Lange, T. (eds.) INDOCRYPT 2006. LNCS, vol. 4329, pp. 148–175. Springer, Heidelberg (2006)
Koblitz, N., Menezes, A.: Another look at “provable security”. Journal of Cryptology 20, 3–37 (2004)
Raddum, H., Semaev, I.: New technique for solving sparse equation systems. Cryptology ePrint Archive, Report 2006/475 (2006), http://eprint.iacr.org/
Sugita, M., Kawazoe, M., Imai, H.: Gröbner basis based cryptanalysis of sha-1. Cryptology ePrint Archive, Report 2006/098(2006), http://eprint.iacr.org/
Wang, X., Yin, Y.L., Yu, H.: Finding Collisions in the Full SHA-1. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 17–36. Springer, Heidelberg (2005)
Wang, X., Yu, H.: How to break md5 and other hash functions. In: Cramer, R.J.F. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 19–35. Springer, Heidelberg (2005)
Wolf, C., Preneel, B.: Taxonomy of public key schemes based on the problem of multivariate quadratic equations. Cryptology ePrint Archive, Report 2005/077, 64 (May, 2005) http://eprint.iacr.org/2005/077/
Yang, B.-Y., Chen, J.-M.: All in the XL Family: Theory and Practice. In: Park, C.-s., Chee, S. (eds.) ICISC 2004. LNCS, vol. 3506, pp. 67–86. Springer, Heidelberg (2005)
Yang, B.-Y., Chen, O.C.-H., Bernstein, D.J., Chen, J.-M.: Analysis of QUAD. In: Biryukov, A. (ed.) Fast Software Encryption — FSE 2007. volume to appear of Lecture Notes in Computer Science, pp. 302–319. Springer, Heidelberg (2007) workshop record available
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Ding, J., Yang, BY. (2008). Multivariates Polynomials for Hashing. In: Pei, D., Yung, M., Lin, D., Wu, C. (eds) Information Security and Cryptology. Inscrypt 2007. Lecture Notes in Computer Science, vol 4990. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79499-8_28
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DOI: https://doi.org/10.1007/978-3-540-79499-8_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79498-1
Online ISBN: 978-3-540-79499-8
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