A generalization of the anomalous attack for the ECDLP over Q _p

The elliptic curve discrete logarithm problem (ECDLP) over a field K is as follows: given an elliptic curve E over K, a point S ∈ E(K), and a point T ∈ E(K) with T ∈ hSi, find the integer d such that T = dS. The hardness of the ECDLP over a finite field is essential for the security of all elliptic curve cryptographic schemes. Semaev, Smart, and Satoh and Araki independently proposed an efficient attack for the ECDLP over F _p in the anomalous case, which is called the anomalous attack. In this paper, we generalize the method of the anomalous attack and give an algorithm for solving the ECDLP over the p-adic field Q _p.