The Elliptic Curve Discrete Logarithm Problems over the p-adic Field and Formal Groups

Abstract

The hardness of the elliptic curve discrete logarithm problem (ECDLP) on a finite field is essential for the security of all elliptic curve cryptographic schemes. The ECDLP on 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 ∈ 〈S 〉, find the integer d such that T = dS. A number of ways of approaching the solution to the ECDLP on a finite field is known, for example, the MOV attack [5], and the anomalous attack [7,10]. In this paper, we propose an algorithm to solve the ECDLP on the p-adic field $\mathcal{Q}_p$. Our method is to use the theory of formal groups associated to elliptic curves, which is used for the anomalous attack proposed by Smart [10], and Satoh and Araki [7].