Public-Key Cryptography

4 min read#cryptography

Split the key in two — a public lock anyone can close and a private key only you can open — using one-way functions with a hidden trapdoor.

Contents

Public-Key Cryptography

For all of history, encryption meant a single shared secret: the same key locked and unlocked, so the two parties had to meet or trust a courier first. Public-key cryptography — also called asymmetric cryptography — shattered that assumption. Each person holds a key pair:

  • a public key, published to the world, that anyone can use to encrypt to you or verify your signature;
  • a private key, kept secret, that only you can use to decrypt or sign.

The metaphor is an open padlock. You hand out copies of an unlocked padlock (your public key); anyone can snap one shut around a message and send it back. Only you hold the physical key that opens it. Closing a padlock takes no secret; opening it does.

Trapdoor one-way functions

The whole edifice stands on a special kind of function: a trapdoor one-way function.

  • One-way — easy to compute forward, infeasible to invert. Like smashing a plate: trivial to do, hopeless to reverse from the pieces.
  • Trapdoor — unless you hold a secret, in which case inversion becomes easy again.

The public key describes the one-way function; the private key is the trapdoor. Two concrete trapdoors power the real world, both living in Modular Arithmetic:

  • Factoring — multiplying primes p \cdot q is easy; factoring n back apart is hard. This is RSA's trapdoor.
  • Discrete logarithm — computing g^a \bmod p is easy; recovering a is hard. This powers Diffie–Hellman and elliptic-curve schemes.
The same operation, two directions. Forward (locking) is cheap for everyone — the bar fills instantly. Backward (unlocking) is a cliff for an attacker without the trapdoor, but instant for the private-key holder. Move the mouse to scale the key size and watch the gap explode.

What the two keys buy you

Asymmetry gives two distinct powers, depending on which key acts first:

  • Encrypt with the public key → decrypt with the private key. Anyone can send you a secret; only you can read it. This is confidentiality.
  • Sign with the private key → verify with the public key. Only you can produce the value; anyone can check it. This is a Digital Signature, giving authenticity and integrity.

In practice asymmetric crypto is slow, so it rarely encrypts bulk data directly. Instead it does the hard part — agreeing on or delivering a short symmetric key (often via Diffie–Hellman) — and a fast symmetric cipher handles the rest. Public-key crypto is the handshake; symmetric crypto is the conversation.

See also

Related topics

  • Diffie–HellmanTwo strangers agree on a shared secret over a fully public channel — the paint-mixing trick that launched public-key cryptography.
  • RSAPublic-key encryption and signatures built on the difficulty of factoring a product of two large primes.
  • Digital SignatureSign a message's hash with your private key so anyone can verify, with your public key, that you wrote it and nobody altered it.
  • Modular ArithmeticArithmetic that wraps around a fixed modulus — the finite-ring algebra that makes public-key cryptography possible.
  • One-Time PadXOR a message with a truly random key as long as itself — the only cipher proven unbreakable, and the only one almost nobody can use.
  • CryptographyThe art and mathematics of keeping secrets and proving identity — from a Roman shift cipher to the public-key algebra that secures the modern web.

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