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RSA: cryptography from number theory

Public keys from two primes, and why factoring is the lock.

Choose primes p, q; n = pq; pick e coprime to φ(n) = (p − 1)(q − 1); the private key d is e⁻¹ mod φ(n). Encrypt mᵉ mod n, decrypt with d. The security is the difficulty of factoring n. Picture it: a padlock anyone can snap shut (e) that only the key (d) opens. Think it: correctness is Euler's theorem; the whole scheme is one modular inverse.

Worked example · inverse of 17 mod 3120

Number-Theory · Modular-Arithmetic

Inverse of 17 mod 3120

17,\ 3120

Step by step

  1. 17x \equiv 1 \pmod{3120}

    We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.

  2. 3120 = 183 \times 17 + 9

    Euclid step.

  3. 17 = 1 \times 9 + 8

    Euclid step.

  4. 9 = 1 \times 8 + 1

    Euclid step.

  5. 8 = 8 \times 1 + 0

    Euclid step.

  6. 17 \times 2753 = 46801 \equiv 1 \pmod{3120}

    Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

Reveal the answer
17^{-1} \equiv 2753 \pmod{3120}

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How to: RSA: cryptography from number theory

  1. We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
  2. Euclid step.
  3. Euclid step.
  4. Euclid step.
  5. Euclid step.
  6. Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller-Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

More in Number Theory

Prime factorisationPrime numbersGCD and LCMModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theorem