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Mathematical Foundations of Cryptography

Role InstructorWhere AzTU, Baku

Visual explainers for the course. Each one runs in the browser: change the inputs, step through the algorithm or the attack, and see exactly where things go right or wrong.

Number theory

Extended Euclid and modular inverses

Watch the remainders shrink while each row keeps r = x·a + y·n. When the remainder hits 1, x is the inverse. Toy RSA gives d.

gcd · Bézout · a⁻¹ mod nOpen →

Chinese Remainder Theorem

Two dials turn together, one mod m and one mod n, until both show the residues you asked for. Then the whole grid lights up.

x ≡ a (mod m) · x ≡ b (mod n)Open →

Powers mod p: Fermat, Euler and orders

Follow a, a², a³, … around a circle until the cycle closes.

aᵖ⁻¹ ≡ 1 · ord(a) | p − 1Coming soon

RSA from key generation to decryption

Pick two primes and follow every number through the scheme.

n = pq · ed ≡ 1 (mod φ)Coming soon

Elliptic curves

Adding points on an elliptic curve

The chord-and-tangent rule over the real numbers, then the same formulas over 𝔽ₚ, where the line wraps around the grid.

y² = x³ + ax + b · P + Q · kGOpen →

Lattices and post-quantum

Good and bad bases of one lattice

The same grid of points from two bases. With a short basis, rounding finds the closest point; with a long, skewed one it fails.

L(B) · CVP · Babai · LWEOpen →

Signatures

ECDSA: reuse a nonce, lose the key

Two signatures with the same k share the same r. Four lines of algebra later, the private key falls out.

s = k⁻¹(z + r·d) mod nOpen →

Applications

Tallying encrypted ballots

Encrypt every ballot, combine the ciphertexts, decrypt only the total. Paillier does it with factoring; an LWE scheme does it with noise.

Enc(a) · Enc(b) = Enc(a + b)Open →