I'm writing a math paper on RSA and it would be nice if it had some calculus in it. Is RSA directly related to calculus in any manner? This can include proving theorems, generating keys, or cracking RSA.
The closest I found is the prime number theorem.
On
Many factoring algorithms, especially the so-called "sieving" algorithms, have a complexity estimate (not a proof, though) based on some analytic results.
One relatively intuitive notion: in order to factor some number $n$, you give yourself an upper bound $B$. Letting $\pi(B)$ be the number of primes up to $B$, the goal is to find about $\pi(B)$ values of a certain form which factor completely with the primes up to $B$ (such a number is called $B$-smooth). The question now is how to choose $B$ in order to make this taks least difficult, but there are conflicting goals. We might be tempted to choose $B$ small, because then we need to find only a small number of suitable values. But then, $B$-smooth numbers are exceedingly rare, so we will have to test a large number of values before we find enough of them. On the other hand, if we choose $B$ large, we will need to find a large number of values, but also it will take longer to determine whether a given value is $B$-smooth.
Using analysis, we can express the time it takes to find $N$ $B$-smooth numbers as a function of $B$ and $N$, and look for a minimum—and we find that the minimum occurs when $\pi(B)$ is about $\exp\left(\sqrt{\log n \log\log n}\right)$.
For more details, see Pomerance's delightful Tale of Two Sieves.
Method 1: AKS algorithm can be used to test primality of a number, and getting large prime number is needed in RSA. Now, in practice, AKS isn't used because it is inefficient at the size of prime we do care about, but of course, it's the best theoretically. The paper that introduce it, Prime is in P refers to a result from another paper, and once you look up the other paper, which is On Chebyshev-type inequalities for primes (this is not truely public access though, but lots of school have access to JStor, and I don't know any other way to access the article), you will find calculus being used, a bit, on polynomial. That is enough pretext for you to cram calculus into your paper.
Method 2: look for prime number theorem, which is a complex analysis result, but close enough to calculus. The excuse to talk about this is that it explains how we will practically never run out of primes number of the size we care about, as the prime number theorem estimate the number of primes and it's in fact reasonably large.