Geometric Ideas for Cryptographic Equation Solving in Even Characteristic

Authors:
Sean Murphy;Maura B. Paterson
Affiliations:
Information Security Group, Dept. of Mathematics, Royal Holloway, University of London, Egham, U.K. TW20 0EX;Information Security Group, Dept. of Mathematics, Royal Holloway, University of London, Egham, U.K. TW20 0EX
Venue:
Cryptography and Coding '09 Proceedings of the 12th IMA International Conference on Cryptography and Coding
Year:
2009

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Abstract

The GeometricXL algorithm is a geometrically invariant version of the XL algorithm that uses polynomials of a much smaller degree than either a standard Groebner basis algorithm or an XL algorithm for certain multivariate equation systems. However, the GeometricXL algorithm as originally described is not well-suited to fields of even characteristic. This paper discusses adaptations of the GeometricXL algorithm to even characteristic, in which the solution to a multivariate system is found by finding a matrix of low rank in the linear span of a collection of matrices. These adaptations of the GeometricXL algorithm, termed the EGHAM process, also use polynomials of a much smaller degree than a Groebner basis or an XL algorithm for certain equation systems. Furthermore, the paper gives a criterion which generally makes a Groebner basis or standard XL algorithm more efficient in many cryptographic situations.