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Question

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

Answer

"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 mod 3 ≡ 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 φ-function
  • Carl Friedrich Gauss (1777–1855): Prime counting function π(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:

Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12.
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12.
The original uploader was Spindled at English Wikipedia. · CC BY-SA 3.0 · Wikimedia Commons
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