From Number Theory to Internet Security

How number theory became a key ingredient in RSA and elliptic-curve cryptography. Real internet security also depends on symmetric ciphers, hashes, protocols, and careful implementation.

About 30 min·8 nodes

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STEP 1 · Concept~BCE 300Alexandria· Publication

Prime Numbers

Integers greater than 1 divisible only by 1 and themselves — the multiplicative atoms of whole numbers. Euclid proved there are infinitely many; primes also support several, but not all, modern cryptographic systems.

↓550-year span

STEP 2 · Mathematician~250Alexandria· Main activity

Diophantus of Alexandria

An ancient author whose life is almost unknown but whose problem-solving voice remains vivid. The usual third-century dates are inferred rather than documented. The Arithmetica collects equations seeking particular rational solutions and uses abbreviations for the unknown and its powers, an important step from rhetorical toward symbolic algebra. Six Greek books survive. An Arabic manuscript found in Mashhad in 1968 identifies itself as four books translated by Qusta ibn Luqa, though scholars debate whether they are lost books of Diophantus or part of a commentary tradition. The puzzle yielding an age of 84 comes from a much later Greek anthology, not a verified tomb inscription.

↓1.4-thousand-year span

STEP 3 · Mathematician1640Toulouse· Main activity

Pierre de Fermat

A pioneer of number theory. Famous for Fermat's Last Theorem — a margin note that took 358 years to prove.

↓160-year span

STEP 4 · Concept1801Göttingen· Publication

Modular Arithmetic

The arithmetic of clocks — wrapping around at a modulus. The basic language of number theory and cryptography.

↓95 years later

STEP 5 · Concept1896Paris· Discovery

Prime Number Theorem

The prime-counting function π(x) is asymptotic to x/ln(x). Gauss recalled an early table-based observation in an 1849 letter; Hadamard and de la Vallée Poussin proved the theorem independently in 1896.

↓81 years later

STEP 6 · Concept1977Cambridge, MA· Discovery

Cryptography and Information

The mathematics of secrets — from Caesar ciphers to quantum key distribution. Information security is fundamentally a question of which math is hard.

↓8 years later

STEP 7 · Concept1985No location

Elliptic Curves

A nonsingular cubic curve whose points carry a group law. Elliptic curves connect ancient questions about rational solutions, nineteenth-century elliptic functions, modern number theory, and selected cryptographic systems.

↓10 years later

STEP 8 · Mathematician1995Princeton· Main activity

Andrew Wiles

The mathematician who proved Fermat's Last Theorem in 1995, ending a 358-year siege through modular forms and elliptic curves.

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