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Geometric Ideas for Cryptographic Equation Solving in Even Characteristic

机译：求解偶数特征的密码方程的几何思想

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摘要

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.
机译：GeometricXL算法是XL算法的几何不变形式，对于某些多元方程组，它使用比标准Groebner基算法或XL算法小得多的次数的多项式。但是，最初描述的GeometricXL算法不适用于偶数特性的字段。本文讨论了GeometricXL算法对偶数特征的适应，其中通过在矩阵集合的线性跨度中找到低秩矩阵来找到多元系统的解决方案。对某些几何方程系统，GeometricXL算法的这些改编（称为EGHAM过程）还使用了比Groebner基或X​​L算法小得多的多项式。此外，本文提供了一种标准，该标准通常使Groebner基础或标准XL算法在许多密码情况下更为有效。

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