Murphy, S. and Paterson, Maura B. (2008) A geometric view of cryptographic equation solving. Journal of Mathematical Cryptology 2 (1), pp. 63-107. ISSN 1862-2976.
2908.pdf - Accepted Version
Download (414Kb) | Preview
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.
|Keyword(s) / Subject(s):||Projective geometry, Veronese variety, determinantal variety, multivariate polynomials, cryptology, linearisation, relinearisation, XL algorithm, geometricXL algorithm|
|School or Research Centre:||Birkbeck Schools and Research Centres > School of Business, Economics & Informatics > Economics, Mathematics and Statistics|
|Date Deposited:||08 Dec 2010 10:10|
|Last Modified:||24 May 2013 07:21|
Archive Staff Only (login required)