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

15$\begingroup$ I'm not sure why this question was closed. Maybe not specific enough? Maybe the question could be phrased as "Could you point me to some literature on the use of elliptic curves over $\mathbb Q$ in cryptography?" But it seems a legitimate question. I'm going to vote to reopen. If you feel it should be closed, please provide a reason. $\endgroup$– Joe SilvermanMay 9, 2019 at 12:19
1 Answer
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, SpringerVerlag, Berlin, 2009, 82102.