Extended F5 criteria

T Faugère's F5 is one of the fastest known algorithm to compute Gröbner bases (see Faugère, 2002). The efficiency of this algorithm comes from two criteria namely F5 criteria, for which it assigns to each polynomial a signature. In this paper, we study the importance of choosing an ordering on the signatures, and we propose a novel ordering on the signatures. Using this ordering, we extend the F5 criteria, and we describe a new algorithm like F5 based on these extended criteria which (despite of F5) does not depend on the order of input polynomials. We have implemented our algorithm in Magma for computing the Gröbner basis of a general ideal, and we evaluate its performance via some examples. We show that the new algorithm is more stable and more efficient than F5, and experimentally it stops at a lower degree than F5.