Relation between the XL Algorithm and Grobner Basis Algorithms

Abstract

We clarify a relation between the XL algorithm and known Gröbner basis algorithms. The XL algorithm was proposed to be a more efficient algorithm to solve a system of algebraic equations under a special condition, without calculating a whole Gröbner basis. But in our result, it is shown that to solve a system of algebraic equations with a special condition under which the XL algorithm works is equivalent to calculate the reduced Gröbner basis of the ideal associated with the system. Moreover we show that the XL algorithm is a Gröbner basis algorithm which can be represented as a redundant variant of a known Gröbner basis algorithm F 4 .

An Analysis of the XSL Algorithm

“…In [15] it is shown how the T method can be interpreted in terms of Buchberger's Gröbner Basis algorithm. The method is further discussed (in the context of the XL2 [9] algorithm) in [2,4], where some doubts are cast on the general applicability of the method.…”

A Public Key Cipher Algorithm Based on Multivariate Cubic Quasigroups (MCQ)

“…x 1 +x 2 x 3 x 2 +x 3 +x 1 x 2 x 3 +x 1 x 2 +x 2 x 3 x 1 +x 2 +x 1 x 2 +x 1 x 3 x 2 +x 1 x 3 x 2 +x 3 +x 1 x 3 x 1 +x 2 +x 3 +x 2 x 3 x 1 +x 2 +x 1 x 2 +x 2 x 3 x 3 +x 1 x 2 x 1 +x 1 x 2 +x 2 x 3 x 1 +x 2 +x 3 +x 1 x 3 x 1 +x 3 +x 1 x 2 +x 1 x 3 x 1 +x 2 +x 1 x 3 x 1 +x 1 x 3 +x 2 x 3 x 1 +x 2 +x 3 +x 1 x 2 x 1 +x 3 +x 1 x 3 +x 2 x 3 x 1 +x 2 +x 2 x 3 x 2 +x 1 x 2 +x 1 x 3 x 1 x 2 +x 1 x 3 +x 2 x 3 x 1 +x 2 +x 1 x 2 +x 1 x 3 +x 2 x 3 x 1 +x 3 +x 1 x 2 x 2 +x 1 x 3 +x 2 x 3 x 2 +x 3 +x 1 x 3 +x 2 x 3 x 1 +x 3 +x 1 x 2 +x 1 x 3 +x 2 x 3 x 1 +x 3 +x 2 x 3 x 3 +x 1 x 2 +x 1 x 3 x 2 +x 3 +x 1 x 2 +x 2 x 3 x 2 +x 3 +x 1 x 2 +x 1 x 3 +x 2 x 3…”

“…This example executes the previous algorithm step by step; 1-Choose randomly two different elements from Table (1); Let  = x 1 x 2 +x 1 x 3 +x 2 x 3 , and  = x 2 +x 3 +x 1 x 2 +x 2 x 3 . 3 such that, for X 2 = (x 4 , x 5 , x 6 ) T the formula A · X 2 + B i gives a valid quasigroup, for this case, there is a 256 vectors. 4-Set CB as the set of all B i founded in previous step; 5-Take a vector B from CB 1+x 1 +x 2 +x 1 x 3 1+x 1 +x 3 1+x 1 +x 3 +x 1 x 2 +x 1 x 3…”

Algebraic Attacks Galore!

“…However, careful analyses of its running time in [20] and [21] make this appear unlikely. In a number of papers the XL Algorithm has been related to the Gröbner basis methods we discuss in the next section (see for instance [5], [11], and [45]). Let us apply it to the following simple example taken from [1].…”

“…If one modifies the XL algorithm such that it proceeds degree by degree, it can be seen as a version of the F4 algorithm, as was shown in [45]. Let us try the F4 algorithm in a small example.…”