So, in encryption theory, a basic principle is that one has an operation that can be computed in a "forward direction" relatively quickly but for which computing in the "reverse direction" is very time-consuming. Multiplying two primes and factoring a composite number into primes is the archetypal example of this.
What is known about the (expected?) time to find an elementary antiderivative to an elementary function that has an elementary antiderivative? What is known about using this as a viable encryption system?
Moreover, what is known about the (expected?) time to find an elementary antiderivative to an elementary differential form that has an elementary antiderivative 1) on $\mathbb{R}^n$ and 2) on a (say, closed) manifold $M^n$ (although some care must be taken with respect to coordinate patches or extending the form to an embedding of the manifold as to what one means by elementary in the latter case)? The problem of gauge (if I'm using the term correctly; I mean the principle that any two antiderivatives differ by an exact form) must also significantly complicate the problem.