Adding on a bit to David Speyers's answer, the **elliptic curve discrete logarithm problem** (ECDLP) has been studied intensively since the 1980s, and in addition to commercial applications, there is a vast research industry that studies the problem and its applications in cryptography. As Speyer says, it is "believed to be hard", indeed the best known general *classical algorithms* take exponential time (which in your setting means a fractional power of $p$ or $e$), as opposed to various sub-exponential running time algorithms for factorization (used for RSA) or for solving the finite field discrete logarithm problem (find $e$ that solves $A\equiv B^e\bmod p$). However, there is a [*quantum algorithm* due to Schor][1] that would solve the ECDLP in (practical) polynomial time if/when a sufficiently large quantum computer is built. If you want to find out about some of the algorithms that are used to solve the ECDLP, you might read about the [*babystep-giantstep method*][2] or [*Pollard's rho method*][3], each of which (in your setting) has a run-time of roughly $\sqrt{p}\approx\sqrt{e}\approx2^{16}$, so would be quite feasible. In cryptographic applications, the $p$ and the $e$ are more like 200 to 1000 bits.


  [1]: https://en.wikipedia.org/wiki/Shor%27s_algorithm
  [2]: https://en.wikipedia.org/wiki/Baby-step_giant-step
  [3]: https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm_for_logarithms