Secrets & Trust

Shift each letter by the key letter under it: one letter is Caesar, a word is Vigenere, a random key as long as the text is the one-time pad. Letter counts break the first; bit counts save the last.

Every number lands on an hour of a clock with m hours, and sums, products, and powers stay there. Squaring takes a 300-bit power in 600 steps; Fermat's theorem tells a prime clock from a composite.

Divide the larger by the smaller, keep the remainder, repeat: the last divisor is the gcd. Run backwards, the divisions write it as ax + my, and when it is 1, x is the inverse of a on the clock.

Pad a message, absorb it word by word in a state that is mixed and chained, and the final state is its fingerprint. Flip one bit and half the digest changes: so a receiver knows nothing moved.

A server keeps a digest, not the password. Unsalted, every common password is already in the attacker's table; a salt makes each record its own puzzle, and stretching makes each guess cost a login.

A block cipher is a keyed permutation of 16-bit blocks: round key, S-box, bit transpose, repeat. Encrypt a picture block by block and codebook mode leaves its shape showing; chaining takes it away.

Alice and Bob each raise a public generator to a private exponent and swap the results; raising what they got to their own exponent gives both one key. Eve saw it all and still needs an exponent.

Two primes make a public modulus and a secret totient; a public exponent gets a private inverse by Euclid. Anyone can raise a message to e, only the owner can undo it, and an attacker must factor n.

Hash the message, raise the digest to the private exponent, send both. Anyone raises the signature to the public exponent and compares with a fresh digest; any changed letter, bit, or key fails.

A certificate is a signature over a name, a key, and a validity window. A chain of them leads from a server's key to a root the verifier trusts, and one failed check anywhere breaks the chain.

Hellos swap key shares, the certificate names the key, a signature over the transcript proves who is talking, and Finished proves the keys. Put Eve in the middle and watch where it stops.

Commit, take a coin, answer. A prover who knows x passes every round, a cheater is caught half the time per round, and rounds faked with no secret verify just the same.