Cryptography
The art and mathematics of keeping secrets and proving identity — from a Roman shift cipher to the public-key algebra that secures the modern web.
Contents
Cryptography
Cryptography is the science of communicating in the presence of adversaries — keeping a message secret from everyone but its intended reader, and proving that it really came from who it claims to. For most of history it was a craft of clever tricks; in the twentieth century it became a branch of mathematics with theorems about what is possible and what is forever out of reach.
Two questions run through everything here:
- Confidentiality — can I scramble a message so only the right person can read it?
- Authenticity — can I prove I wrote it, and that nobody changed a single bit?
The journey runs from ciphers a child can break to algorithms whose security rests on problems no computer can solve in the lifetime of the universe.
Two eras: symmetric and public-key
For thousands of years all cryptography was symmetric: sender and receiver shared one secret key, used to both lock and unlock. The Caesar Cipher, the Substitution Cipher, and the unbreakable One-Time Pad all live here. The catch is brutal — before you can talk secretly, you must already have shared a secret, somehow.
In the 1970s came the revolution. Public-Key Cryptography split the key in two: a public lock anyone can snap shut, and a private key only you can open. Suddenly strangers could exchange secrets over a tapped wire. Diffie–Hellman lets two parties conjure a shared key in plain sight; RSA turns the difficulty of factoring large numbers into a padlock. All of it is built on the quiet algebra of Modular Arithmetic.
The threads back to information and probability
Cryptography is not an island — it borrows its deepest ideas from the rest of this field.
- Entropy and perfect secrecy. Shannon proved that a cipher is perfectly secret only when the key carries at least as much entropy as the message. That single inequality explains why the One-Time Pad is unbreakable and why every shorter key is, in principle, attackable.
- Hamming Distance and the avalanche. A good Hash Function must turn a one-bit change in its input into a change in roughly half the output bits — a large Hamming distance from a tiny nudge. Avalanche is what makes a fingerprint trustworthy.
- Modular Arithmetic as the engine. Wrap-around arithmetic on a finite ring of integers is the playground where one-way functions live: easy to compute forward, ruinously hard to reverse.
The pages in this section
- Caesar Cipher — shift every letter by a fixed amount; 25 keys, broken in seconds.
- Substitution Cipher — any letter-to-letter map; an astronomical keyspace undone by frequency analysis.
- One-Time Pad — XOR with a truly random key; provably perfect secrecy, hopelessly impractical.
- Modular Arithmetic — clock arithmetic, the algebra beneath all of public-key crypto.
- Diffie–Hellman — agree on a shared secret over a public channel.
- RSA — encryption and signatures from the hardness of factoring.
- Public-Key Cryptography — the big idea: a public lock, a private key.
- Hash Function — a one-way fingerprint with the avalanche property.
- Digital Signature — sign with your private key, verify with your public one.