On the discrete logarithm problem in elliptic curves II

We continue our study on the elliptic curve discrete logarithm problem over finite extension fields. We show, among others, the following results: For sequences of prime powers (qi)(i is an element of N) and natural numbers (ni)(i is an element of N) with n(i) -> infinity and n(i)/log(q(i))(2) -> 0 for i -> infinity , the discrete logarithm problem in the groups of rational points of elliptic curves over the fields F q n i i can be solved in subexponential expected time (q(i)(ni))(o(1)). Let a, b > 0 be fixed. Then the problem over fields F-qn, where q is a prime power and n a natural number with a . log (q)(1/3) <= n <= b . log (q), can be solved in an expected time of e O. log. q n / 3 = 4 /.