Cryptography sometimes uses elliptic curves over finite fields. Does cryptography also use elliptic curves over $\mathbb{Q}$ or rational points on them? 



Cryptography and elliptic curves
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.


