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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.
مثال کار شده · inverse of 17 mod 3120
قدم به قدم
- 17x \equiv 1 \pmod{3120}
We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
- 3120 = 183 \times 17 + 9
Euclid step.
- 17 = 1 \times 9 + 8
Euclid step.
- 9 = 1 \times 8 + 1
Euclid step.
- 8 = 8 \times 1 + 0
Euclid step.
- 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.
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چطور: RSA: cryptography from number theory
- We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
- Euclid step.
- Euclid step.
- Euclid step.
- Euclid step.
- Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.
سوالاتي که مردم ميپرسن
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.
بیشتر در Number Theory
Prime factorisationPrime numbersGCD and LCMModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theorem