Comparison Between XL and Gröbner Basis Algorithms
نویسندگان
چکیده
This paper compares the XL algorithm with known Gröbner basis algorithms. We show that to solve a system of algebraic equations via the XL algorithm is equivalent to calculate the reduced Gröbner basis of the ideal associated with the system. Moreover we show that the XL algorithm is also a Gröbner basis algorithm which can be represented as a redundant variant of a Gröbner basis algorithm F4. Then we compare these algorithms on semi-regular sequences, which correspond, in conjecture, to almost all polynomial systems in two cases: over the fields F2 and Fq with q n. We show that the size of the matrix constructed by XL is large compared to the ones of the F5 algorithm. Finally, we give an experimental study between XL and the Buchberger algorithm on the cryptosystem HFE and find that the Buchberger algorithm has a better
منابع مشابه
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 ...
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