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∈N and natural numbers (ni)i∈N with ni −→ ∞ and ni log(qi) −→ 0 for i −→ ∞, the discrete logarithm problem in the groups of rational points of elliptic curves over the fields Fqi i can be solved in subexponential expected time (qi i ) . Let a, b > 0 be fixed. Then the problem over fields Fqn , where q is a prime power and n a natural number with a · log(q) ≤ n ≤ b · log(q), can be solved in an expected time of e n)3/4).

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