IJPAM: Volume 77, No. 1 (2012)

A GENERALIZATION OF THE ANOMALOUS ATTACK
FOR THE ECDLP OVER $\Q_p$

Masaya Yasuda
Fujitsu Laboratories Ltd.
1-1, Kamikodanaka 4-chome, Nakahara-ku, Kawasaki
211-8588, JAPAN


Abstract. The elliptic curve discrete logarithm problem (ECDLP) over a field $K$ is as follows: given an elliptic curve $E$ over $K$, a point $S \in E(K)$, and a point $T \in E(K)$ with $T \in \langle S \rangle$, 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 $\Q_p$.

Received: May 16, 2011

AMS Subject Classification: 14G52, 11G07

Key Words and Phrases: ECDLP, formal groups, the anomalous attack

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 77
Issue: 1