Relation between the XL Algorithm and Gröbner Basis Algorithms

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 F4. key words: algebraic attacks, Gröbner basis algorithm, F4, XL algorithm