Sean Murphy and Maura Paterson (2007) A Geometric View of Cryptographic Equation Solving.
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This paper considers the geometric properties of the Relinearisation algorithm and of the XL algorithm used in cryptology for equation solving. We give a formal description of each algorithm in terms of projective geometry, making particular use of the Veronese variety. We establish the fundamental geometrical connection between the two algorithms and show how both algorithms can be viewed as being equivalent to the problem of finding a matrix of low rank in the linear span of a collection of matrices, a problem sometimes known as the MinRank problem. Furthermore, we generalise the XL algorithm to a geometrically invariant algorithm, which we term the GeometricXL algorithm. The GeometricXL algorithm is a technique which can solve certain equation systems that are not easily soluble by the XL algorithm or by Groebner basis methods.
This is a Published version This version's date is: 14/05/2007 This item is peer reviewed
https://repository.royalholloway.ac.uk/items/c9e13031-43e5-f425-eab4-4ab63150ac14/1/
Deposited by () on 28-Jun-2010 in Royal Holloway Research Online.Last modified on 14-Dec-2010
1. G. Bard and N. Courtois. Algebraic Cryptanalysis of the Data Encryption Stan-dard , 2006. http://eprint.iacr.org/2006/402.
2. E. Bertini. Introduzione Alla Geometrica Proiettiva Degli Iperspazi, Con Appen-dice Sulle Curve Algebriche E Loro Singolarit¶a. Pisa, 1907. http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=00%790002&seq=5.
3. B. Buchberger. Ein Algorithmus zum Au±nden der Basiselemente des Restk-lassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, UniversitÄatInnsbruck, 1965.
4. E. Carlini. Reducing the Number of Variables of a Ploynomial. In M. Elkadi,B. Mourrain, and R. Piene, editors, Algebraic Geometry and Geometric Modelling(Mathematics and Visualisation), pages 237{247. Springer, 2006.
5. Rey Casse. Projective Geometry: An Introduction. Oxford University Press, 2006.
6. C. Cid, S. Murphy, and M. Robshaw. Algebraic Aspects of the Advanced EncryptionStandard. Springer, 2006.
7. P. Cohn. Classical Algebra. John Wiley, 2000.
8. N. Courtois, A. Klimov, J. Patarin, and A. Shamir. E±cient Algorithms for SolvingOverde¯ned Systems of Multivariate Polynomial Equations. In B. Preneel, editor,Advances in Cryptology { EUROCRYPT 2000, volume 1807 of LNCS, pages 392{407. Springer{Verlag, 2000.
9. D. Cox, J. Little, and D. O'Shea. Ideals, Varieties, and Algorithms. UndergraduateTexts in Mathematics. Springer{Verlag, second edition, 1997.10. D. Cox, J. Little, and D. O'Shea. Using Algebraic Geometry, volume 185 of Grad-uate Texts in Mathematics. Springer, second edition, 2004.
11. C. Diem. The XL-Algorithm and a Conjecture from Commutative Algebra. In P.J.Lee, editor, Advances in Cryptology { ASIACRYPT 2004, volume 3329 of LNCS,pages 323{337. Springer{Verlag, 2004.
12. J-C. Faugµere. A New E±cient Algorithm for Computing GrÄobner bases (F4).Journal of Pure and Applied Algebra, 139:61{88, 1999.
13. J-C. Faugµere. A New E±cient Algorithm for Computing GrÄobner Bases withoutReduction to Zero (F5). In T. Mora, editor, International Symposium on Symbolicand Algebraic Computation { ISSAC 2002, pages 75{83, 2002.
14. W.T. Gowers. How to Lose your Fear of Tensor Products, 2005. http://www.dpmms.cam.ac.uk/\~wtg10/tensors3.html.
15. W. GrÄobner. ÄUber Veronesesche VarietÄaten und deren Projektionen. Arch. Math.,16:257{264, 1965.
16. J. Harris. Algebraic Geometry: A First Course. Number 133 in Graduate Text inMathematics. Springer, 1992.
17. J.W.P. Hirschfeld and J.A. Thas. General Galois Geometries. Oxford UniversityPress, 1991.
18. A. Iarrobino and V. Kanev. Power Sums, Gorenstein Algebras and DeterminantalLoci. Number 1725 in Lecture Notes in Mathematics. Springer, 1999.
19. V. Kanev. Chrodal Varieties of Veronese Varieties and Catalechticant Matrices.Journal of Mathematical Sciences, 94:1114{1125, 1999.
20. A. Kipnis and A. Shamir. Cryptanalysis of the HFE Public Key Cryptosystem byRelinearization. In H. Imai and Y. Zheng, editors, Advances in Cryptology, Crypto'99, volume 1267 of LNCS, pages 19{30. Springer{Verlag, 1999.
21. R. Lidl and H. Niederreiter. Introduction to Finite Fields and Their Applications.Cambridge University Press, 1994.