# Geometric ideas for cryptographic equation solving in even characteristic

Murphy, S. and Paterson, Maura B.
(2009)
Geometric ideas for cryptographic equation solving in even characteristic.
In:
Parker, M.G. (ed.)
*Cryptography and Coding.*
Lecture Notes in Computer Science 5921.
Berlin, Germany:
Springer, pp. 202-221.
ISBN 9783642108679.

Text
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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.

## Metadata

Item Type: | Book Section |
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Additional Information: | 12th IMA International Conference, Cryptography and Coding 2009, Cirencester, UK, December 15-17, 2009. Proceedings |

School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |

Depositing User: | Maura Paterson |

Date Deposited: | 11 Jan 2013 15:51 |

Last Modified: | 09 Aug 2023 12:32 |

URI: | https://eprints.bbk.ac.uk/id/eprint/5352 |

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