On the First Fall degree of the equations system obtained by Weil descent

In this paper, we will study the properties of the curve, which is obtained by Weil descent from a global polynomial. Generally, it is difficult to estimate the upper bound of the first fall degree of the equations system obtained by Weil descent. For a global polynomials F→ and m0→ , which satisfies that m0→ has sufficiently large degree of each variables Xi→ and deg Xi→ F→・m0→ is of the form pα + 1, the upper bound of the first fall degree of the equations system obtained by Weil descent from F→・m0→ can be estimated. Moreover, we see that taking suitable m0→ , the set of the zeros of the equations system obtained by Weil descent of F→ equals to the that of F→・m0→ . Thus we give a refinement of the results of Petit et al. and show that the discrete logarithm problem of elliptic curve over small characteristic field Fpn is subexponential of input size n, under first fall degree assumption and not assuming other heuristics of the properties of global polynomials