Trapdoor one-way permutations and multivariate polynomials
This article is divided into three parts. The first part describes the known candidates of trapdoor one-way permutations. The second part presents a new algorithm, called D*. As we will see, this algorithm is not secure. However, in the third part, D* will be a useful tool to present our new candidate trapdoor one-way permutation, called D**. This candidate is based on properties of multivariate polynomials on finite fields, and has similar characteristics to T. Matsumoto and H. Imai's schemes.
What makes trapdoor one-way permutations particularly interesting is the fact that they immediately provide ciphering, signature, and authentication asymmetric schemes.
Our candidate performs excellently in secret key, and secret key computations can be implemented in low-cost smart-cards, i.e. without co-processors. An extended version of this paper can be obtained from the authors.
KeywordsElliptic Curve Finite Field Elliptic Curf Total Degree Quadratic Residue
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- 1.D. Bleichenbacher, W. Bosma, A.K. Lenstra, Some Remarks on Lucas-Based Cryptosystems, Advances in Cryptology, Proceedings of CRYPTO'95, Springer-Verlag, pp. 386–396.Google Scholar
- 2.I. Blake, X. Gao, R. Mullin, S. Vanstone, T. Yaghoobian, Applications of finite Fields, Kluwer Academic Publishers, p. 25.Google Scholar
- 3.N. Courtois, Les cryptosystémes asymétriques à représentation obscure, Rapport de stage de DEA, Bull PTS — Université Paris 6, 1997. (This paper is available from J. Patarin and L. Goubin).Google Scholar
- 4.N. Demytko, A New Elliptic Curve Based Analogue of RSA, Advances in Cryptology, Proceedings of EUROCRYPT'93, Springer-Verlag, pp. 40–49.Google Scholar
- 5.M. Dickerson, The functional Decomposition of Polynomials, Ph.D Thesis, TR 89-1023, Department of Computer Science, Cornell University, Ithaca, NY, July 1989.Google Scholar
- 6.M. Garey, D. Johnson, Computers and Intractability, a Guide to the Theory of NP-Completeness, Freeman, p. 251.Google Scholar
- 7.K. Koyama, U.M. Maurer, T. Okamoto, S.A. Vanstone, New Public-Key Schemes Based on Elliptic Curves over the Ring Z n, Advances in Cryptology, Proceedings of CRYPTO'91, Springer-Verlag, pp. 252–266.Google Scholar
- 9.R. Lidl, G.L. Mullen, G. Turwald, Pitman Monographs and Surveys in Pure and Applied Mathematics 65: Dickson Polynomials, London, Longman Scientific and Technical, 1993.Google Scholar
- 10.R. Lidl, W.B. Müller, Permutation Polynomials in RSA-Cryptosystems, Advances in Cryptology, Proceedings of CRYPTO'83, Plenum Press, 1984, pp. 293–301.Google Scholar
- 11.T. Matsumoto, H. Imai, Algebraic Methods for Constructing Asymmetric Cryptosystems, AAECC-3, Grenoble, 1985.Google Scholar
- 12.T. Matsumoto, H. Imai, Public Quadratic Polynomial-Tuples for Efficient Signature-Verification and Message-Encryption, Advances in Cryptology, Proceedings of EUROCRYPT'88, Springer-Verlag, pp. 419–453.Google Scholar
- 13.W.B. Müller, Polynomial Functions in Modern Cryptology, Contributions to General Algebra 3: Proceedings of the Vienna Conference, Vienna: Verlag Holder-Pichler-Tempsky, 1985, pp. 7–32.Google Scholar
- 14.W.B. Müller, R. Nöbauer, Some Remarks on Public-Key Cryptography, Studia Scientiarum Mathematicarum Hungarica, v.16, 1981, pp. 71–76.Google Scholar
- 15.W.B. Müller, R. Nöbauer, Cryptanalysis of the Dickson-scheme, Advances in Cryptology, Proceedings of EUROCRYPT'85, Springer-Verlag, pp. 50–61.Google Scholar
- 16.J. Patarin, Cryptanalysis of the Matsumoto and Imai Public Key Scheme of Eurocrypt'88, Advances in Cryptology, Proceedings of CRYPTO'95, Springer-Verlag, pp. 248–261.Google Scholar
- 17.J. Patarin, Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP) Two New Families of Asymmetric Algorithms, Advances in Cryptology, Proceedings of EUROCRYPT'96, Springer-Verlag, pp. 33–48.Google Scholar
- 18.J. Patarin, Asymmetric Cryptography with a Hidden Monomial, Advances in Cryptology, Proceedings of CRYPTO'96, Springer-Verlag, pp. 45–60.Google Scholar
- 19.J. Patarin, L. Goubin, Asymmetric Cryptography with S-boxes, ICICS'97 (this conference.Google Scholar
- 20.M.O. Rabin, Digitized Signatures and Public-Key Functions as Intractable as Factorization, Technical Report LCS/TR-212, M.I.T. Laboratory for Computer Science, 1979.Google Scholar
- 22.P.J. Smith, LUC Public-Key Encryption, Dr. Dobb's Journal, January 1993, pp. 44–49.Google Scholar
- 23.P.J. Smith, M.J.J. Lennon, LUC. a New Public Key System, Proceedings of the Ninth IFIP Int. Symp. on Computer Security, 1993, pp. 103–117.Google Scholar
- 25.H.C. Williams, An M 3 Public-Key Encryption Scheme, Advances in Cryptology, Proceedings of CRYPTO'85, Springer-Verlag, pp. 358–368.Google Scholar