How prime numbers and modular arithmetic protect the digital world.
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The RSA cryptosystem (Rivest-Shamir-Adleman):
p = 61, q = 53. n = 3233, φ(n) = 3120
e = 17 (coprime to 3120). d = 2753 (since 17 × 2753 = 46801 ≡ 1 mod 3120)
Encrypt m = 65: c = 65¹⁷ mod 3233 = 2790
Decrypt: 2790²⁷⁵³ mod 3233 = 65 ✓
Security relies on: factoring n = pq is hard when p, q are ~1024 bits each. Euler's theorem guarantees decryption works: m^(ed) ≡ m^(1 + kφ(n)) ≡ m (mod n).
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