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Overview of Cryptanalysis Techniques in Multivariate Public Key Cryptography

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Gröbner Bases, Coding, and Cryptography

Abstract

This paper summarizes most of the main developments in the cryptanalysis of multivariate cryptosystems and discuss some problems that remain open. A strong emphasis is put on the symbolic computation tools that have been used to achieve these advances.

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References

  • G. Ars, J. C. Faugère, H. Imai, M. Kawazoe, and M. Sugita, Comparison between XL and Gröbner basis algorithms, Proc. of Asiacrypt 2004 (P. J. Lee, ed.), LNCS, vol. 3329, Springer, Berlin, 2004, pp. 338–353.

    Google Scholar 

  • M. Bardet, An investigation on overdetermined algebraic systems and applications to error-correcting codes and to cryptography, Ph.D. thesis, University of Paris 6, Paris, France, 2004.

    Google Scholar 

  • C. Berbain and H. Gilbert, On the security of IV dependent stream ciphers, FSE 2007 (A. Biryukov, ed.), LNCS, vol. 4593, Springer, Berlin, 2007, pp. 254–273.

    Google Scholar 

  • C. Berbain, H. Gilbert, and J. Patarin, QUAD: A practical stream cipher with provable security, EUROCRYPT 2006 (S. Vaudenay, ed.), LNCS, vol. 4004, Springer, Berlin, 2006, pp. 109–128.

    Google Scholar 

  • O. Billet and H. Gilbert, A traceable block cipher, Asiacrypt 2003 (C. S. Laih, ed.), LNCS, vol. 2894, Springer, Berlin, 2003, pp. 331–346.

    Google Scholar 

  • O. Billet and H. Gilbert, Cryptanalysis of Rainbow, SCN 2006 (R. De Prisco and M. Yung, eds.), LNCS, vol. 4116, Springer, Berlin, 2006, pp. 336–347.

    Google Scholar 

  • O. Billet, M. J. B. Robshaw, and T. Peyrin, On building hash functions from multivariate quadratic equations, ACISP 2007 (J. Pieprzyk, H. Ghodosi and E. Dawson, eds.), LNCS, vol. 4586, Springer, Berlin, 2007, pp. 82–95.

    Google Scholar 

  • O. Billet, J. Patarin, and Y. Seurin, Analysis of Intermediate Field Systems, SCC 2008 (D. Wang and J.-C. Faugère, eds.), 2008.

    Google Scholar 

  • A. Biryukov, B. Preneel, A. Braeken, and C. de Cannière, A toolbox for cryptanalysis: linear and affine equivalence algorithms, Eurocrypt 2003 (E. Biham, ed.), LNCS, vol. 2656, Springer, Berlin, 2003, pp. 33–50.

    Google Scholar 

  • A. Braeken, B. Preneel, and C. Wolf, A study of the security of unbalanced Oil & Vinegar signature schemes, CT-RSA 2005 (A. Menezes, ed.), LNCS, vol. 3376, 2005, p. 29.

    Google Scholar 

  • B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, Innsbruck, 1965.

    Google Scholar 

  • B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math. 4 (1970), 374–383.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Buchberger, Gröbner-bases: An algorithmic method in polynomial ideal theory, Multidimensional systems theory, Reidel, Dordrecht, 1985, pp. 184–232.

    Chapter  Google Scholar 

  • B. Buchberger, An algorithmical criterion for the solvability of algebraic systems of equations, London Math. Soc. LNS 251 (1998), 535–545.

    MathSciNet  Google Scholar 

  • B. Buchberger, Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, J. Symb. Comput. 41 (2006), nos. 3–4, 475–511.

    Article  MathSciNet  MATH  Google Scholar 

  • J. F. Buss, G. S. Frandsen, and J. O. Shallit, The computational complexity of some problems of linear algebra, J. Comput. Syst. Sci. 58 (1999), no. 3, 572–596.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Caboara, F. Caruso, and C. Traverso, Gröbner bases for public key cryptography, Proc. of ISSAC 2008 (L. Gonzalez-Vega, ed.), ACM, New York, 2008.

    Google Scholar 

  • D. Coppersmith, J. Stern, and S. Vaudenay, Attacks on the birational permutation signature schemes, CRYPTO93 (D. R. Stinson, ed.), LNCS, vol. 773, Springer, Berlin, 1993, pp. 435–443.

    Google Scholar 

  • D. Coppersmith, J. Stern, and S. Vaudenay, The security of the birational permutation signature schemes, Journal of Cryptology 10 (1997), no. 3, 207–221.

    Article  MathSciNet  MATH  Google Scholar 

  • N. T. Courtois, The security of Hidden Field Equations (HFE), Proc. of CT-RSA 2001 (D. Naccache, ed.), LNCS, vol. 2020, Springer, Berlin, 2001, pp. 266–281.

    Google Scholar 

  • N. Courtois and W. Meier, Algebraic attacks on stream ciphers with linear feedback, EUROCRYPT 2003 (E. Biham, ed.), LNCS, vol. 2656, Springer, Berlin, 2003, pp. 345–359.

    Google Scholar 

  • N. Courtois, A. Klimov, J. Patarin, and A. Shamir, Efficient algorithms for solving overdefined systems of multivariate polynomial equations, Proc. of EUROCRYPT 2000, LNCS, vol. 1807, Springer, Berlin, 2000, pp. 392–407.

    Google Scholar 

  • M. T. Dickerson, The functional decomposition of polynomials, Ph.D. thesis, Cornell University, Ithaca, NY, USA, 1989.

    Google Scholar 

  • L. E. Dickson, History of the theory of numbers, vol. 3, Chelsea, New York, 1971.

    Google Scholar 

  • J. Ding and J. E. Gower, Inoculating multivariate schemes against differential attacks, Cryptology ePrint Archive, Report 2005/255, 2005.

    Google Scholar 

  • J. Ding and D. Schmidt, A defect of the implementation schemes of the TTM cryptosystem, Cryptology ePrint Archive, Report 2003/085, 2003.

    Google Scholar 

  • J. Ding and D. Schmidt, The new implementation schemes of the TTM cryptosystem are not secure, Progr. Comput. Sci. Appl. Logic 23 (2004), 113–127.

    MathSciNet  Google Scholar 

  • J. Ding and D. Schmidt, Rainbow, a new multivariable polynomial signature scheme, ACNS 2005 (J. Ioannidis, A. D. Keromytis and M. Yung, eds.), LNCS, vol. 3531, Springer, Berlin, 2005a, pp. 164–175.

    Google Scholar 

  • J. Ding and D. Schmidt, Cryptanalysis of HVEv and internal perturbation of HFE, PKC 2005 (S. Vaudenay, ed.), LNCS, vol. 3386, Springer, Berlin, 2005b, p. 288.

    Google Scholar 

  • J. Ding, L. Hu, X. Nie, J. Li, and J. Wagner, High order linearization equation (HOLE) attack on multivariate public key cryptosystems, PKC 2007 (T. Okamoto and X. Wang, eds.), LNCS, Springer, Berlin, 2007a.

    Google Scholar 

  • J. Ding, C. Wolf, and B.-Y. Yang, ℓ -invertible cycles for multivariate quadratic (MQ) public key cryptography, PKC 2007 (T. Okamoto and X. Wang, eds.), LNCS, vol. 4450, Springer, Berlin, 2007b, pp. 266–281.

    Google Scholar 

  • V. Dubois, P.-A. Fouque, A. Shamir, and J. Stern, Practical cryptanalysis of SFLASH, CRYPTO 2007 (A. Menezes, ed.), LNCS, vol. 4622, Springer, Berlin, 2007a, pp. 1–12.

    Google Scholar 

  • V. Dubois, L. Granboulan, and J. Stern, Cryptanalysis of HFE with internal perturbation, PKC 2007 (T. Okamoto and X. Wang, eds.), LNCS, vol. 3494, Springer, Berlin, 2007b.

    Google Scholar 

  • J. C. Faugére, A new efficient algorithm for computing Gröbner bases (F 4), J. Pure Appl. Algebra 139 (1999), nos. 1–3, 61–88.

    Article  MathSciNet  MATH  Google Scholar 

  • J. C. Faugère, A new efficient algorithm for computing Gröbner bases without reduction to zero (F 5), Proc. of ISSAC 2002, ACM, New York, 2002, pp. 75–83.

    Google Scholar 

  • J. Faugère and G. Ars, An algebraic cryptanalysis of nonlinear filter generators using Gröbner bases, INRIA Research Report 4739, 2003.

    Google Scholar 

  • J. C. Faugère and A. Joux, Algebraic cryptanalysis of hidden field equation (HFE) cryptosystems using Gröbner bases, LNCS, vol. 2729 Springer, Berlin, 2003, pp. 44–60.

    Google Scholar 

  • J. C. Faugère and L. Perret, Polynomial equivalence problems: algorithmic and theoretical aspects, EUROCRYPT 2006, LNCS, vol. 4004, Springer, Berlin, 2006a, pp. 30–47.

    Google Scholar 

  • J. C. Faugère and L. Perret, Cryptanalysis of 2R \(^{\mbox{-}}\) schemes, CRYPTO 2006 (C. Dwork, ed.), LNCS, vol. 4117, Springer, Berlin, 2006b, pp. 357–372.

    Google Scholar 

  • J.-C. Faugère and L. Perret, Cryptanalysis of MinRank, CRYPTO 2008 (D. Wagner, ed.), LNCS, vol. 5157, Springer, Berlin, 2008a, pp. 280–296.

    Google Scholar 

  • J.-C. Faugère and L. Perret, On the security of UOV, SCC 2008 (D. Wang and J. C. Faugère, eds.), 2008b.

    Google Scholar 

  • H. J. Fell and W. Diffie, Analysis of a public key approach based on polynomial substitution, CRYPTO 85 (H. C. Williams, ed.), LNCS, vol. 218, Springer, Berlin, 1985, pp. 340–349.

    Google Scholar 

  • M. Fellows and N. Koblitz, Combinatorial cryptosystems galore!, Finite Fields: Theory, Applications, and Algorithms (G. L. Mullen and P. J.-S. Shiue, eds.), Contemporary Mathematics, vol. 168, AMS, Providence, 1994, pp. 51–61.

    Chapter  Google Scholar 

  • P.-A. Fouque, L. Granboulan, and J. Stern, Differential cryptanalysis for multivariate schemes, EUROCRYPT 2005 (R. Cramer, ed.), LNCS, vol. 3494, Springer, Berlin, 2005, pp. 341–353.

    Google Scholar 

  • P. A. Fouque, G. Macario-Rat, L. Perret, and J. Stern, Total break of the -IC signature scheme, PKC 2008, LNCS, vol. 4939, Springer, Berlin, 2008a, pp. 1–17.

    Google Scholar 

  • P.-A. Fouque, G. Macario-Rat, and J. Stern, Key recovery on hidden monomial multivariate schemes, EUROCRYPT 2008 (N. P. Smart, ed.), LNCS, vol. 4965, Springer, Berlin, 2008b, pp. 19–30.

    Google Scholar 

  • A. S. Fraenkel and Y. Yesha, Complexity of solving algebraic equations, Inf. Process. Lett. 10 (1980), nos. 4–5, 178–179.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Geiselmann, R. Steinwandt, and T. Beth, Attacking the affine parts of SFLASH, Cryptography and coding—IMA 2001, Springer, Berlin, 2001, pp. 355–359.

    Chapter  Google Scholar 

  • L. Goubin, Théorie et Pratique de la Cryptologie sur Carte à Microprocesseur, Mémoire d’habilitation à diriger des recherches, 2003.

    Google Scholar 

  • L. Goubin and N. T. Courtois, Cryptanalysis of the TTM cryptosystem, ASIACRYPT 2000 (T. Okamoto, ed.), LNCS, vol. 1976, Springer, Berlin, 2000, pp. 44–57.

    Google Scholar 

  • L. Granboulan, A. Joux, and J. Stern, Inverting HFE is quasipolynomial, CRYPTO2006 (C. Dwork, ed.), LNCS, vol. 4117, Springer, Berlin, 2006, pp. 345–356.

    Google Scholar 

  • H. Imai and T. Matsumoto, Algebraic methods for constructing asymmetric cryptosystems, Proc. of AAECC 3, LNCS, vol. 229, Springer, Berlin, 1985, pp. 108–119.

    Google Scholar 

  • X. Jiang, J. Ding, and L. Hu, Kipnis-Shamir’s attack on HFE revisited, Inscrypt 2007 (D. Feng and Y. Zhang, eds.), LNCS, Springer, Berlin, 2007.

    Google Scholar 

  • A. Kipnis and A. Shamir, Cryptanalysis of the oil & vinegar signature scheme, CRYPTO ’98, LNCS, vol. 1462, Springer, Berlin, 1998, pp. 257–266.

    Google Scholar 

  • A. Kipnis and A. Shamir, Cryptanalysis of the HFE public key cryptosystem by relinearization, CRYPTO 99 (M. J. Wiener, ed.), LNCS, vol. 1666, Springer, Berlin, 1999, pp. 19–30.

    Google Scholar 

  • A. Kipnis, J. Patarin, and L. Goubin, Unbalanced oil & vinegar signature schemes, EUROCRYPT ’99 (J. Stern, ed.), LNCS, vol. 1592, Springer, Berlin, 1999, pp. 206–222.

    Google Scholar 

  • D. E. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third ed., Addison–Wesley, Reading, 1997.

    Google Scholar 

  • N. Koblitz, Algebraic aspects of cryptography, Algorithms and computation in mathematics, vol. 3, Springer, Berlin, 1999.

    Google Scholar 

  • F. Levy-dit-Vehel, M. G. Marinari, L. Perret, and C. Traverso, A survey on Polly Cracker systems, this volume, 2009, pp. 285–305.

    Google Scholar 

  • F. S. Macaulay, The algebraic theory of modular systems, Cambridge University Press, Cambridge, 1916.

    MATH  Google Scholar 

  • R. J. McEliece, A public key cryptosystem based on algebraic coding theory, JPL DSN 42–44 (1978), 114–116.

    Google Scholar 

  • T. T. Moh, A fast public key system with signature and master key functions, Proc. of CrypTEC99, Hong Kong City Press, 1999.

    Google Scholar 

  • T. Mora, Gröbner technology, this volume, 2009, pp. 11–25.

    Google Scholar 

  • P. Q. Nguyen and J. Stern, The two faces of lattices in cryptology, CaLC 2001 (J. H. Silverman, ed.), LNCS, vol. 2146, Springer, Berlin, 2001, pp. 146–180.

    Google Scholar 

  • H. Niederreiter, Knapsack-type cryptosystems and algebraic coding theory, Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform. 15 (1986), no. 2, 159–166.

    MathSciNet  MATH  Google Scholar 

  • J. Patarin, Cryptoanalysis of the Matsumoto and Imai public key scheme of Eurocrypt ’88, CRYPTO 95 (D. Coppersmith, ed.), LNCS, vol. 963, Springer, Berlin, 1995, pp. 248–261.

    Google Scholar 

  • J. Patarin, Hidden fields equations (HFE) and isomorphisms of polynomials (IP): two new families of asymmetric algorithms, EUROCRYPT ’96 (U. M. Maurer, ed.), LNCS, vol. 1070, Springer, Berlin, 1996, pp. 33–48.

    Google Scholar 

  • J. Patarin, The oil & vinegar signature scheme, Proc. of Dagstuhl Workshop on Cryptography, 1997.

    Google Scholar 

  • J. Patarin, Challenge HFE, http://www.minrank.org/hfe#challenge, 1998.

  • J. Patarin and L. Goubin, Asymmetric cryptography with S-boxes, ICICS 97, LNCS, vol. 1334, Springer, Berlin, 1997, pp. 369–380.

    Google Scholar 

  • J. Patarin, L. Goubin, and N. T. Courtois, C *−+ and HM: variations around two schemes of T. Matsumoto and H. Imai, ASIACRYPT ’98 (K. Ohta and D. Pei, eds.), LNCS, vol. 1514, Springer, Berlin, 1998a, pp. 35–49.

    Google Scholar 

  • J. Patarin, L. Goubin, and N. T. Courtois, Improved algorithms for isomorphisms of polynomials, EUROCRYPT 98 (K. Nyberg, ed.), LNCS, vol. 1403, Springer, Berlin, 1998b, pp. 184–200.

    Google Scholar 

  • J. Patarin, L. Goubin, and N. T. Courtois, SFLASH, a Fast Asymmetric Signature Scheme for Low Cost Smart-Cards, https://www.cosic.esat.kuleuven.be/nessie/workshop/submissions/sflash.zip, 2000.

  • L. Perret, A fast cryptanalysis of the isomorphism of polynomials with one secret problem, EUROCRYPT 2005 (R. Cramer, ed.), LNCS, vol. 3494, Springer, Berlin, 2005, pp. 354–370.

    Google Scholar 

  • O. Regev, Lattice-based cryptography, Proc. of CRYPTO2006 (C. Dwork, ed.), LNCS, vol. 4117, Springer, Berlin, 2006, pp. 131–141.

    Google Scholar 

  • A. Shamir, Efficient signature schemes based on birational permutations, CRYPTO93 (D. R. Stinson, ed.), LNCS, vol. 773, Springer, Berlin, 1993, pp. 1–12.

    Google Scholar 

  • P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26 (1997), 1484–1509.

    Article  MathSciNet  MATH  Google Scholar 

  • T. Thierauf, The computational complexity of equivalence and isomorphism problems, LNCS, vol. 1852, Springer, Berlin, 2000, pp. 1–135.

    Chapter  Google Scholar 

  • J. von zur Gathen and V. Shoup, Computing Frobenius Maps and Factoring Polynomials, Computational Complexity 2 (1992), 187–224.

    Article  MathSciNet  MATH  Google Scholar 

  • J. von zur Gathen, J. Gutierrez, and R. Rubio, Multivariate polynomial decomposition, Appl. Algebra Eng. Commun. Comput. 14 (2003), no. 1, 11–31.

    MATH  Google Scholar 

  • L.-C. Wang, B.-Y. Yang, Y.-H. Hu, and F. Lai, A “Medium-Field” multivariate public-key encryption scheme, CT-RSA 2006 (D. Pointcheval, ed.), LNCS, vol. 3860, Springer, Berlin, 2006, pp. 132–149.

    Google Scholar 

  • B.-Y. Yang and J.-M. Chen, Building secure tame-like multivariate public-key cryptosystems: The new TTS, ACISP 2005 (C. Boyd and J. M. G. Nieto, eds.), LNCS, vol. 3574, Springer, Berlin, 2005, pp. 518–531.

    Google Scholar 

  • B.-Y. Yang, J.-M. Chen, and Y.-H. Chen, TTS: High-speed signatures on a low-cost smart card, CHES 2004 (M. Joye and J. J. Quisquater, eds.), LNCS, vol. 3156, Springer, Berlin, 2004, pp. 371–385.

    Google Scholar 

  • D.-F. Ye, K.-Y. Lam, and Z.-D. Dai, Cryptanalysis of “2 R” schemes, CRYPTO 99, LNCS, vol. 1666, Springer, Berlin, 1999, pp. 315–325.

    Google Scholar 

  • D.-F. Ye, Z.-D. Dai, and K.-Y. Lam, Decomposing attacks on asymmetric cryptography based on mapping compositions, J. of Cryptology 14 (2001), no. 2, 137–150.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Orange Labs, 38–40 rue du Général Leclerc, 92794, Issy-les-Moulineaux, France

    Olivier Billet

  2. Department of Mathematical Sciences, Department of Computer Sciences, University of Cincinnati, Cincinnati, OH, 45220, USA

    Jintai Ding

Authors
  1. Olivier Billet
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Correspondence to Olivier Billet .

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Editors and Affiliations

  1. Boole Centre for Research in Informatics, University College Cork, Crosses Green 17 South Bank, Cork, Ireland

    Massimiliano Sala

  2. Faculty of Engineering Dept. Production Systems Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka Toyohashi, Aichi, Tempaku-cho, 441-8580, Japan

    Shojiro Sakata

  3. Dipto. Matematica, Università Genova, Via Dodecaneso, 35, 16146, Genova, Italy

    Teo Mora

  4. Dipto. Matematica, Università Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy

    Carlo Traverso

  5. LIP6, University of Paris VI, 104 avenue du Président Kennedy, 75016, Paris, France

    Ludovic Perret

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Billet, O., Ding, J. (2009). Overview of Cryptanalysis Techniques in Multivariate Public Key Cryptography. In: Sala, M., Sakata, S., Mora, T., Traverso, C., Perret, L. (eds) Gröbner Bases, Coding, and Cryptography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-93806-4_15

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