Cryptanalysis of multivariate and Odd-characteristic HFE variants

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 6571 LNCS, Issue , 2011, Pages 441-458

  Bettale, Luk ^a   Faugère, Jean Charles ^a   Perret, Ludovic ^a  

^a UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

Author keywords

Gr bner bases; Hidden Field Equations; MinRank

Indexed keywords

EXTENSION FIELD; HIDDEN FIELD EQUATIONS; KEY RECOVERY ATTACKS; MINRANK; PROOF OF CONCEPT; PUBLIC KEYS;

SECURITY OF DATA;

PUBLIC KEY CRYPTOGRAPHY;

EID: 79952495417     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-642-19379-8_27     Document Type: Conference Paper

References (31)

1. Adams, W.W., Loustaunau, P.: An Introduction to Gröbner Bases. Graduate Studies in Mahematics, vol. 3. AMS, Providence (1994)
2. Bardet, M., Faug̀ere, J.-C., Salvy, B.: On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In: Proc. International Conference on Polynomial System Solving (ICPSS), pp. 71-75 (2004)
3. Bardet, M., Faug̀ere, J.-C., Salvy, B., Yang, B.-Y.: Asymptotic behaviour of the degree of regularity of semi-regular polynomial systems. In: Proc. of MEGA 2005, Eighth International Symposium on Effective Methods in Algebraic Geometry (2005)
4. Bettale, L., Faug̀ere, J.-C., Perret, L.: Hybrid approach for solving multivariate systems over finite fields. Journal of Mathematical Cryptology, 177-197 (2009)
5. Billet, O., Patarin, J., Seurin, Y.: Analysis of Intermediate Field Systems. In: SCC 2008 (2008)
6. Bogdanov, A., Eisenbarth, T., Rupp, A., Wolf, C.: Time-area optimized publickey engines: MQ-cryptosystems as replacement for elliptic curves? In: Oswald, E., Rohatgi, P. (eds.) CHES 2008. LNCS, vol. 5154, pp. 45-61. Springer, Heidelberg (2008)
7. Bosma, W., Cannon, J.J., Playoust, C.: The Magma algebra system I: The user language. Journal of Symbolic Computation 24(3-4), 235-265 (1997) (Pubitemid 127167874)
8. Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph.D. thesis, University of Innsbruck (1965)
9. Buss, W., Frandsen, G., Shallit, J.: The computational complexity of some problems of linear algebra. Journal of Computer and System Sciences (1999)
10. Chen, A.I.-T., Chen, M.-S., Chen, T.-R., Cheng, C.-M., Ding, J., Kuo, E.L.-H., Lee, F.Y.-S., Yang, B.-Y.: SSE implementation of multivariate PKCs on modern x86 CPUs. In: Clavier, C., Gaj, K. (eds.) CHES 2009. LNCS, vol. 5747, pp. 33-48. Springer, Heidelberg (2009)
11. Chen, C.H.O., Chen, M.S., Ding, J., Werner, F., Yang, B.Y.: Odd-char multivariate Hidden Field Equations. Cryptology ePrint Archive (2008), http://eprint.iacr.org/2008/543
12. Courtois, N.T.: Efficient zero-knowledge authentication based on a linear algebra problem MinRank. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 402-421. Springer, Heidelberg (2001) (Pubitemid 33371198)
13. Cox, D.A., Little, J.B., O'Shea, D.: Ideals, Varieties and Algorithms. Springer, Heidelberg (2005)
14. Ding, J., Schmidt, D., Werner, F.: Algebraic attack on HFE revisited. In: Wu, T.-C., Lei, C.-L., Rijmen, V., Lee, D.-T. (eds.) ISC 2008. LNCS, vol. 5222, pp. 215-227. Springer, Heidelberg (2008)
15. Faug̀ere, J.-C.: A new efficient algorithm for computing Gröbner bases (F4). Journal of Pure and Applied Algebra 139, 61-88 (1999)
16. Faug̀ere, J.-C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation ISSAC, pp. 75-83. ACM Press, New York (2002) (Pubitemid 35023374)
17. Faug̀ere, J.-C.: Algebraic cryptanalysis of HFE using Gröbner bases. Reasearch report RR-4738, INRIA (2003), http://hal.inria.fr/inria- 00071849/PDF/RR-4738.pdf
18. Faug̀ere, J.-C.: FGb: A Library for Computing Gröbner Bases. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 84-87. Springer, Heidelberg (2010), http://www-salsa.lip6.fr/ ~jcf/Papers/ICMS.pdf
19. Faug̀ere, J.-C., Joux, A.: Algebraic cryptanalysis of Hidden Field Equation (HFE) cryptosystems using Gröbner bases. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 44-60. Springer, Heidelberg (2003) (Pubitemid 137636934)
20. Faug̀ere, J.-C., Levy-dit-Vehel, F., Perret, L.: Cryptanalysis of MinRank. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 280-296. Springer, Heidelberg (2008)
21. Faug̀ere, J.-C., Safey El Din, M., Spaenlehauer, P.J.: Computing loci of rank defects of linear matrices using Gröbner bases and applications to cryptology. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation 2010 - ISSAC 2010 (2010)
22. Faug̀ere, J.C., Safey El Din, M., Spaenlehauer, P.J.: Gröbner Bases of Bihomogeneous Ideals Generated by Polynomials of Bidegree (1,1): Algorithms and Complexity. Journal of Symbolic Computation, 1-39 (2010)
23. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
24. Granboulan, L., Joux, A., Stern, J.: Inverting HFE is quasipolynomial. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 345-356. Springer, Heidelberg (2006) (Pubitemid 44532125)
25. Jiang, X., Ding, J., Hu, L.: Kipnis-Shamir attack on HFE revisited. In: Pei, D., Yung, M., Lin, D., Wu, C. (eds.) Inscrypt 2007. LNCS, vol. 4990, pp. 399-411. Springer, Heidelberg (2008)
26. Kipnis, A., Shamir, A.: Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 19-30. Springer, Heidelberg (1999)
27. Matsumoto, T., Imai, H.: Public quadratic polynomial-tuples for efficient signatureverification and message-encryption. In: Günther, C.G. (ed.) EUROCRYPT 1988. LNCS, vol. 330, pp. 419-453. Springer, Heidelberg (1988)
28. Patarin, J.: Cryptoanalysis of the Matsumoto and Imai public key scheme of Eurocrypt 1988. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 248-261. Springer, Heidelberg (1995)
29. 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) (Pubitemid 126071052)
30. Wolf, C., Preneel, B.: Equivalent keys in HFE, C *, and variations. In: Dawson, E., Vaudenay, S. (eds.) Mycrypt 2005. LNCS, vol. 3715, pp. 33-49. Springer, Heidelberg (2005)
31. Wolf, C., Preneel, B.: Large superfluous keys in multivariate quadratic asymmetric systems. In: Vaudenay, S. (ed.) PKC 2005. LNCS, vol. 3386, pp. 275-287. Springer, Heidelberg (2005) (Pubitemid 41231337)