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    A geometric view of cryptographic equation solving

    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.

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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.


    Item Type: Article
    Keyword(s) / Subject(s): Projective geometry, Veronese variety, determinantal variety, multivariate polynomials, cryptology, linearisation, relinearisation, XL algorithm, geometricXL algorithm
    School: Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences
    Depositing User: Administrator
    Date Deposited: 08 Dec 2010 10:10
    Last Modified: 09 Aug 2023 12:30


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