Cryptology provides a real-world application of many topics in number theory: integer factorization, primality testing, quadratic reciprocity, and elliptic curves, just to name a few. For these applications to cryptology, it is important to know whether or not a given procedure can be performed quickly. How does one convey to students that, for example, primality testing is relatively fast while integer factorization is relatively slow? We will present labs designed for UC Irvine Math 173AB: Introduction to Cryptology. These labs use Sage to introduce relevant cryptology topics, and at the same time they enable students to work with numbers at the limits of what their computers can handle computationally. For such numbers, the difference between a "fast" algorithm and a "slow" algorithm is striking, and as a result, students learn a key principle justifying the security of many modern cryptosystems.