A geometric view of cryptographic equation solving

S. Murphy 1 , 2  and M. B. Paterson 1 , 2
  • 1 Information Security Group, Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, U.K. Email: s.murphy@rhul.ac.uk
  • 2 Information Security Group, Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, U.K. Email: m.b.paterson@rhul.ac.uk

Abstract

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.

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JMC is a forum for original research articles in the area of mathematical cryptology. Works in the theory of cryptology and articles linking mathematics with cryptology are welcome. Submissions from all areas of mathematics significant for cryptology are published, including but not limited to, algebra, algebraic geometry, coding theory, combinatorics, number theory, probability and stochastic processes.

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