We study the elliptic curve discrete logarithm problem over finite extension fields. We show that for any sequences of prime powers (qi)i∈ℕ and natural numbers (ni)i∈ℕ with ni⟶∞ and ni/log (qi)⟶0 for i⟶∞, the elliptic curve discrete logarithm problem restricted to curves over the fields 𝔽qnii can be solved in subexponential expected time (qnii)o(1). We also show that there exists a sequence of prime powers (qi)i∈ℕ such that the problem restricted to curves over 𝔽qi can be solved in an expected time of e𝒪(log (qi)2/3).
