About the XL algorithm over GF(2)

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

Volumn 2612, Issue , 2003, Pages 141-157

  Courtois, Nicolas T ^a   Patarin, Jacques ^a,b  

^a CP8 Crypto Lab   (France)

^b UNIVERSITY OF VERSAILLES   (France)

Author keywords

Gr bner bases; MQ problem; Multivariate quadratic equations; Overdefined and exactly defined systems of multivariate equations; XL algorithm; XSL attacks on AES

Indexed keywords

ALGORITHMS; PROBLEM SOLVING; PUBLIC KEY CRYPTOGRAPHY;

ALGORITHM FOR SOLVING; LINEARLY INDEPENDENTS; MQ PROBLEM; MULTIVARIATE EQUATIONS; MULTIVARIATE QUADRATIC EQUATIONS; PUBLIC KEY CRYPTOSYSTEMS; QUADRATIC EQUATIONS; XSL ATTACK;

CRYPTOGRAPHY;

EID: 35248834096     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/3-540-36563-x_10     Document Type: Article

References (15)

1. B. Barkee, D. C. Can, J. Ecks, T. Moriarty, R. F. Ree: Why You Cannot Even Hope to use Gröbner Bases in Public Key Cryptography: An Open Letter to a Scientist Who Failed and a Challenge to Those Who Have Not Yet Failed, in Journal of Symbolic Computation 18, 1994, S. 497-501.
2. Don Coppersmith, Shmuel Winograd: "Matrix multiplication via arithmetic progressions"; J. Symbolic Computation (1990), 9, pp. 251-280.
3. Nicolas Courtois, Louis Goubin, Willi Meier, Jean-Daniel Tacier: Solving under-defined Systems of Multivariate Quadratic Equations; PKC 2002, LNCS 2274, Springer, pp. 211-227.
4. Nicolas Courtois: The security of Hidden Field Equations (HFE); Cryptographers' Track Rsa Conference 2001, San Francisco 8-12 April 2001, LNCS2020, Springer-Verlag, pp. 266-281.
5. Nicolas Courtois and Josef Pieprzyk, Cryptanalysis of Block Ciphers with Overdefined Systems of Equations; to be presented at Asiacrypt 2002, a preprint with a different version of the attack is available at http://eprint.iacr.org/2002/044/.
6. Magnus Daum: Das Kryptosystem HFE und quadratische Gleichungssysteme über endlichen Körpern, Diplomarbeit, Universität Dortmund, 2001. Available from daum@itsc.ruhr-uni-bochum.de.
7. Jean-Charles Faugère: Computing Gröbner basis without reduction to 0, , technical report LIP6, in preparation, source: private communication.
8. Also presented at the Workshop on Applications of Commutative Algebra, Catania, Italy, 3-6 April 2002.
9. Jean-Charles Faugère: Report on a successful attack of HFE Challege 1 with Gröbner bases algorithm F5/2, announcement that appeared in sci.crypt newsgroup on the internet in April 19th 2002.
10. Michael Garey, David Johnson: Computers and Intractability, a guide to the theory of NP-completeness, Freeman, p. 251.
11. Mireille Martin-Deschamps, private communication.
12. T. T. Moh: On The Method of XL and Its Inefficiency Against TTM, available at http://eprint.iacr.org/2001/047/.
13. Jacques Patarin: Hidden Fields Equations (HFE) and Isomorphisms of Polynomials (IP): two new families of Asymmetric Algorithms; in Eurocrypt'96, Springer Verlag, pp. 33-48.
14. Adi Shamir, Aviad Kipnis: Cryptanalysis of the HFE Public Key Cryptosystem; In Advances in Cryptology, Proceedings of Crypto'99, Springer-Verlag, LNCS.
15. Adi Shamir, Jacques Patarin, Nicolas Courtois, Alexander Klimov, Efficient Algorithms for solving Overdefined Systems of Multivariate Polynomial Equations, Eurocrypt'2000, LNCS 1807, Springer, pp. 392-407.