Paper 2016/1189

On the Bit Security of Elliptic Curve Diffie--Hellman

Barak Shani

Abstract

This paper gives the first bit security result for the elliptic curve Diffie--Hellman key exchange protocol for elliptic curves defined over prime fields. About $5/6$ of the most significant bits of the $x$-coordinate of the Diffie--Hellman key are as hard to compute as the entire key. A similar result can be derived for the $5/6$ lower bits. The paper also generalizes and improves the result for elliptic curves over extension fields, that shows that computing one component (in the ground field) of the Diffie--Hellman key is as hard to compute as the entire key.