TITLE: Cryptography and elliptic curves
QUESTION [20 upvotes]: Cryptography sometimes uses elliptic curves over finite fields. Does cryptography also use elliptic curves over $\mathbb{Q}$ or rational points on them?

REPLY [31 votes]: Not directly, as far as I know, since explicitly computing large multiples of points in $E(\mathbb Q)$ is infeasible. However, people have considered lifting points from $E(\mathbb F_p)$ to $E(\mathbb Q)$ or to the $p$-adics $E(\mathbb Q_p)$ in order to devise algorithms to solve the discrete log problem in $E(\mathbb F_p)$ (although, unsuccessfully so far). Here are a few papers to get you started:

Elliptic curve discrete logarithms and the index calculus. Advances in cryptology—ASIACRYPT'98 (Beijing), 110–125, Lecture Notes in Comput. Sci., 1514, Springer, Berlin, 1998
The xedni calculus and the elliptic curve discrete logarithm problem. Des. Codes Cryptogr. 20 (2000), no. 1, 5–40;  Analysis of the xedni calculus attack. Des. Codes Cryptogr. 20 (2000), no. 1, 41–64. 
Lifting and elliptic curve discrete logarithms, Selected Areas of
Cryptography (SAC 2008), Lecture Notes in Computer Science 5381,
Springer-Verlag, Berlin, 2009, 82--102.