← Cipher Museum Hall of Foundations
The Mathematics Under the Cryptography

The ground your
keys stand on

Every modern cipher in the museum rests on mathematics far older than any protocol. This hall shows that bedrock — the number theory, the hardness assumptions, the structures — with a working interactive for each, and an honest account of where the math stops being something a cryptographer directly uses.

What belongs here. These are foundations, not primitives. You stand on every one of them when you generate a key or verify a signature — but you never call them directly. For the deployable tools, see Modern Cryptography and Hall XI. This hall is for understanding why they work.

The number theory ECC & RSA rest on

The arithmetic world where public-key cryptography actually happens.

§142

Finite Fields

𝔽ₚ · arithmetic mod a prime

Watch a smooth elliptic curve shatter into a scatter of dots — and see why the discrete log becomes hard.

 §143

Modular Arithmetic

ℤ/nℤ · clock arithmetic

Primitive roots and the cyclic group Diffie–Hellman lives inside.

 §144

The Prime Number Theorem

π(x) ~ x / ln x

Why RSA can find a 2048-bit prime in moments — the one theorem here that's genuinely load-bearing.

 §145

Euler & Fermat

aᵠ⁽ⁿ⁾ ≡ 1 (mod n)

The little theorem that makes RSA decryption undo RSA encryption. Watch a message round-trip.

Hardness assumptions — the honest bedrock

Cryptography rests on problems believed hard. Here is what that means.

§146

The One-Way Function

easy → · hard ←

The conceptual keystone everything is downstream of. Multiply in a blink; factor and sweat.

 §147

Lattices & the SVP

ℤⁿ · shortest vector

Drag a basis and hunt the shortest vector — the foundation under Kyber, Dilithium, and the post-quantum batch.

 §152

The Closest Vector Problem

CVP · LWE · Kyber's core

Round a target to the nearest lattice corner — right with a short basis, wrong with a skewed one. This is decoding.

 §148

Reduction & Proof

A ≤ B · "as hard as"

What "provably as hard as factoring" actually claims — and what it quietly assumes.

Structures the protocols assume

The vocabulary half the modern wing silently depends on.

§149

Group Theory

⟨g⟩ · cyclic groups & order

Generators, order, and cyclic structure — the grammar of the discrete-log family.

 §150

Elliptic Curves over ℝ

y² = x³ + ax + b

A gentle first look at the group law, drawn by hand, before it moves to a finite field.

 §151

Information & Entropy

H = −Σ p log p

Shannon's measure of surprise: why the one-time pad is perfect and why key length is destiny.

Post-quantum & advanced structures

The exact math the deployed standards and privacy protocols are built from.

§153

Polynomial Rings

R_q = ℤ_q[X]/(Xⁿ+1)

The structured lattices inside ML-KEM and ML-DSA — watch polynomials wrap around the ring boundary.

 §154

The Random Oracle

avalanche · Fiat–Shamir

Flip one input bit, scramble half the output — the idealized hash assumption beneath Fiat–Shamir signatures.

 §155

Polynomial Interpolation

k points fix a degree-(k−1) curve

Drag points until the curve locks and the secret intercept appears — the heart of Shamir sharing.

 §156

Bilinear Pairings

e(aP,bQ) = e(P,Q)ᵃᵇ

Verify a product of hidden scalars without revealing them — the magic under zk-SNARKs.

 §157

Learning With Errors

t = As + e

Solve for the secret, then switch the noise on and watch the same algebra fail — the structure ML-KEM actually holds.

Companion exhibit: The Ground Beneath the Curve — the modularity theorem and the number theory ECC inherits. Start there if you came from the Fermat's Last Theorem thread.