What is the central insight behind the mathematics of asymmetric cryptography?

"Centuries-old mathematics in the most modern applications." — The math behind RSA, Diffie-Hellman, and ECC was developed hundreds of years ago by Fermat, Euler, Gauss, and others.

The key operations are surprisingly simple — they all boil down to modular arithmetic, which is essentially the integer remainder after division. As one description puts it: "In primary school you learned 20 ÷ 3 = 6 remainder 2. At university, we make it even simpler: we only care about the remainder. So $20 \bmod 3 \equiv 2$."

The six mathematical "protagonists" whose work underlies all of modern asymmetric crypto:

• Pierre de Fermat (1607–1665): Little Fermat's Theorem
• Leonhard Euler (1707–1783): Euler's Theorem, Euler's $\varphi$-function
• Carl Friedrich Gauss (1777–1855): Prime counting function $\pi(n)$
• Niels Henrik Abel (1802–1829): Abelian (commutative) groups
• Évariste Galois (1811–1832): Group theory, Galois fields
• George Boole (1815–1864): Boolean algebra, XOR operation

Go deeper:

• Modular arithmetic (Wikipedia) — the single operation every asymmetric scheme reduces to.