Mathematics
Primitive Roots and Discrete Logarithms in Finite Fields
Quick fact
The multiplicative group of a finite field is always cyclic: there exists an element whose powers produce every nonzero element. For a prime p, the number of such primitive roots is φ(p−1), and no efficient classical algorithm is known for computing discrete logarithms, which is why they're used in cryptography.
Why this is interesting
You know multiplication and division. But what if you had an operation where going forward is easy, yet going backward is so hard that it keeps your online messages safe? That's the world of primitive roots and discrete logarithms.