On asymptotic security estimates in XL and gröbner bases-related algebraic cryptanalysis

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

Volumn 3269, Issue , 2004, Pages 401-413

  Yang, Bo Yin ^a   Chen, Jiun Ming ^a   Courtois, Nicolas T ^a  

^a NONE

Author keywords

Algebraic cryptanalysis; Asymptotic security estimates; Gr bner Bases; Multivariate quadratics; XL

Indexed keywords

ALGEBRA; POLYNOMIALS;

ALGEBRAIC ATTACK; ALGEBRAIC CRYPTANALYSIS; ASYMPTOTIC FORMULA; ASYMPTOTIC SECURITY ESTIMATES; MULTIVARIATE POLYNOMIAL; MULTIVARIATE QUADRATICS; NP-HARD; UNIFIED APPROACH;

CRYPTOGRAPHY;

EID: 33646811417     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/978-3-540-30191-2_31     Document Type: Article

References (42)

1. Anonymous Referee Report from Crypto 2004.
2. M. Bardet, J.-C. Faugère, and B. Salvy, Complexity of Gröbner Basis Computations for Regular Overdetermined Systems, INRIA RR-5049.
3. D. Bernstein, Matrix Inversion Made Difficult, preprint at http://cr.yp.to.
4. C. Chester, B. Friedman, and F. Ursell, An Extension of the Method of Steepest Descents, Proc. Camb. Philo. Soc. 53 (1957) pp. 599-611.
5. D. Coppersmith, private communication.
6. D. Coppersmith, S. Winograd, Matrix multiplication via Arithmetic Progressions, J. Symbolic Computation, 9 (1990), pp. 251-280.
7. N. Courtois, Higher-Order Correlation Attacks, XL Algorithm and Cryptanalysis of Toyocrypt, ICISC '02, LNCS v. 2587, pp. 182-199.
8. N. Courtois, Fast Algebraic Attacks on Stream Ciphers with Linear Feedback, CRYPTO'03, LNCS v. 2729, pp. 177-194.
9. v2, PKC '04, LNCS v. 2947, pp. 201-217.
10. v3, a Fast Asymmetric Signature Scheme, preprint available at http://eprint.iacr.org/2003/ 211.
11. N. Courtois, A. Klimov, J. Patarin, and A. Shamir, Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations, EUROCRYPT 2000, LNCS v. 1807, pp. 392-407.
12. N. Courtois and J. Patarin, About the XL Algorithm over GF(2), CT-RSA 2003, LNCS v. 2612, pp. 141-157.
13. N. Courtois and J. Pieprzyk, Cryptanalysis of Block Ciphers with Overdefined Systems of Equations, ASIACRYPT 2002, LNCS v. 2501, pp. 267-287.
14. J. Daemen and V. Rijmen, The Design of Rijndael, AES - The Advanced Encryption Standard. Springer-Verlag, 2002.
15. C. Diem, The XL-algorithm and a conjecture from commutative algebra, ASIACRYPT 2004, to appear.
16. I. S. Duff, A. M. Erismann, and J. K. Reid, Direct Methods for Sparse Matrices, published by Oxford Science Publications, 1986.
17. W. Eberly and E. Kaltofen, On Randomized Lanczos Algorithms, Proc. ISSAC '97, pp. 176-183, ACM Press 1997.
18. D. Eisenbud, Commutative Algebra with a View toward Algebraic Geometry, Springer-Verlag 1995.
19. J.-C. Faugère, A New Efficient Algorithm for Computing Gröbner Bases without Reduction to Zero (F5), Proceedings of ISSAC 2002, pp. 75-83, ACM Press 2002.
20. J.-C. Faugère and A. Joux, Algebraic Cryptanalysis of Hidden Field Equations (HFE) Cryptosystems Using Gröbner Bases, CRYPTO 2003, LNCS v. 2729, pp. 44-60.
21. R. Fröberg, An inequality for Hubert Series of Graded Algebras, Math. Scand. 56(1985) 117-144.
22. M. Garey and D. Johnson, Computers and Intractability, A Guide to the Theory of NP-completeness, W. H. Freeman New York 1979.
23. Hsien-Kuei Hwang, Asymptotic estimates of elementary probability distributions, Studies in Applied Mathematics, 99:4 (1997), pp. 393-417.
24. B. LaMacchia and A. Odlyzko, Solving Large Sparse Linear Systems over Finite Fields, CRYPTO'90, LNCS V. 537, pp. 109-133.
25. D. Lazard, Gröbner Bases, Gaussian Elimination and Resolution of Systems of Algebraic Equations, EUROCAL '83, LNCS V. 162, pp. 146-156.
26. T. Matsumoto and H. Imai, Public Quadratic Polynomial-Tuples for Efficient Signature-Verification and Message-Encryption, EUROCRYPT'88, LNCS V. 330, pp. 419-453.
27. C. McGeoch, "Veni, Divisi, Vici", Appearing in the "Computer Science Sampler" column of the Amer. Math. Monthly, May 1995.
28. T. Moh, On The Method of XL and Its Inefficiency Against TTM, available at http://eprint.iacr.org/2001/047
29. S. Murphy and M. Robshaw, Essential Algebraic Structures Within the AES, CRYPTO 2002, LNCS V. 2442, pp. 1-16.
30. S. Murphy and M. Robshaw, Comments on the Security of the AES and the XSL Technique, from author's homepage http://www.isg.rhul.ac.uk/~sean/
31. NESSIE Security Report, V2.0, available at http://www.cryptonessie.org
32. J. Patarin, Hidden Field Equations (HFE) and Isomorphisms of Polynomials (IP): Two New Families of Asymmetric Algorithms, EUROCRYPT'96, LNCS V. 1070, pp. 33-48.
33. -+ and HM: Variations Around Two Schemes of T. Matsumoto and H. Imai, ASIACRYPT'98, LNCS V. 1514, pp. 35-49.
34. v2 available at http://www.cryptonessie.org
35. V. Strassen, Gaussian Elimination is not Optimal, Num. Math. 13(1969) pp. 354-356.
36. M. Sugita, M. Kawazoe, and H. Imai, Relation between XL algorithm and Groebner Bases Algorithms, preprint, http://eprint.iacr.org/2004/112.
37. G. Szegö, Orthogonal Polynomials, 4th ed., publ.: Amer. Math. Soc., Providence.
38. D. Wiedemann, Solving Sparse Linear Equations over Finite Fields, IEEE Transaction on Information Theory, v. IT-32 (1976), no. 1, pp. 54-62.
39. R. Wong, Asymptotic Approximations of Integrals, Acad. Press (San Diego) 1989.
40. B.-Y. Yang and J.-M. Chen, All in the XL Family: Theory and Practice, preprint.
41. B.-Y. Yang and J.-M. Chen, TTS: Rank Attacks in Tame-Like Multivariate PKCs, available at http://eprint.iacr.org/2004/061.
42. B.-Y. Yang and J.-M. Chen, Theoretical Analysis of XL over Small Fields, ACISP 2004, LNCS V. 3108, pp. 277-288. Note: updated version available from the authors.