The elliptic curve discrete logarithm problems over the p-adic field and formal groups

Authors:
Masaya Yasuda
Affiliations:
Fujitsu Laboratories Ltd., Kawasaki, Japan
Venue:
ISPEC'10 Proceedings of the 6th international conference on Information Security Practice and Experience
Year:
2010

Citing 4
Cited 0

Use of elliptic curves in cryptography

Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p

Mathematics of Computation
Elliptic curves in cryptography

Elliptic curves in cryptography
Guide to Elliptic Curve Cryptography

Guide to Elliptic Curve Cryptography

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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].