Learn the Math

? What is Modular Arithmetic?

Modular arithmetic is like clock math. When the hour hand goes past 12, it wraps back to 1. In math terms, we're working with remainders after division. This simple concept is the foundation of nearly all modern encryption.

For example: 17 mod 5 = 2 (because 17 ÷ 5 = 3 remainder 2)

★ Key Properties

• Addition: (a + b) mod n = ((a mod n) + (b mod n)) mod n
• Multiplication: (a × b) mod n = ((a mod n) × (b mod n)) mod n
• Exponentiation: Used in RSA encryption for secure key generation
• One-way function: Easy to compute forward, nearly impossible to reverse

▶ Try It Yourself

? What are Prime Numbers?

A prime number is a natural number greater than 1 that can only be divided evenly by 1 and itself. These seemingly simple numbers are the building blocks of all integers and form the foundation of cryptographic security.

The beauty of primes: every integer greater than 1 can be uniquely expressed as a product of prime numbers (Fundamental Theorem of Arithmetic).

★ Why Primes Matter in Cryptography

• RSA Encryption: Uses the product of two large primes (1024+ bits each)
• One-way trap: Multiplying primes is easy; factoring their product is extremely hard
• Key generation: Your private keys depend on prime number properties
• Security guarantee: No known efficient algorithm can factor large semiprimes

▶ Try It Yourself

? What are Discrete Logarithms?

A discrete logarithm is the inverse of modular exponentiation. If g^x mod p = h, then x is the discrete logarithm of h to the base g, modulo p. The magic: computing g^x mod p is easy, but finding x from the result is incredibly hard.

This asymmetry (easy one way, hard the other) is what makes public-key cryptography possible.

★ The Diffie-Hellman Key Exchange

• Step 1: Alice and Bob agree on public values g and p
• Step 2: Alice picks secret a, sends g^a mod p to Bob
• Step 3: Bob picks secret b, sends g^b mod p to Alice
• Step 4: Both compute (g^ab mod p) - the shared secret!

Even if an attacker sees all public communication, they cannot compute the shared secret without solving the discrete log problem.

▶ Try It Yourself

? What are Elliptic Curves?

An elliptic curve is a special type of curve defined by the equation y² = x³ + ax + b. When we combine this with modular arithmetic over a prime field, we get Elliptic Curve Cryptography (ECC) - the backbone of Bitcoin and Ethereum.

ECC provides the same security as RSA with much smaller key sizes: a 256-bit ECC key ≈ 3072-bit RSA key.

★ Point Operations on Curves

• Point Addition: Given two points P and Q, we can compute P + Q geometrically
• Point Doubling: Computing P + P = 2P using the tangent line
• Scalar Multiplication: Computing kP by repeated doubling and adding
• ECDLP: Given P and Q = kP, finding k is computationally infeasible

⛓ secp256k1: Bitcoin's Curve

Bitcoin uses a specific elliptic curve called secp256k1, defined by:

Where p is a massive 256-bit prime. Your Bitcoin private key is a random 256-bit number k, and your public key is the point K = k × G, where G is the generator point.

⚛ Why today’s encryption is in danger

RSA and elliptic curves are safe today because reversing them (factoring huge numbers or solving discrete logs) would take normal computers billions of years. But a large quantum computer running Shor’s algorithm could do it in hours. That is the whole reason post-quantum cryptography exists.

▦ Lattices: hard problems in a grid of dots

Most post-quantum schemes are built on lattices — endless grids of points made by adding and subtracting a few vectors. The security comes from problems that are easy to state but brutally hard to solve, even for quantum computers.

• Shortest Vector Problem (SVP): in a huge grid of points, find the point closest to the origin. Easy to picture, incredibly hard to compute.
• Learning With Errors (LWE): hide a secret inside many equations, then sprinkle in tiny random noise so the equations can’t be solved.

🛡 The new standards

In 2024, the US standards body NIST published the first official post-quantum algorithms. These are already rolling out in browsers and messaging apps right now.

• Kyber (ML-KEM): a lattice-based key exchange — how two devices agree on a secret quantum-safely.
• Dilithium (ML-DSA): lattice-based digital signatures that prove who sent a message.
• SPHINCS+ (SLH-DSA): signatures built only from hash functions — a totally different backup approach.

? What is symmetric encryption?

In symmetric cryptography a single secret key is used to both lock (encrypt) and unlock (decrypt) a message. It is far faster than public-key math, so almost all real data — your Wi-Fi, disk encryption, and the actual content of an HTTPS page — is protected with it. Public-key crypto is mostly used just to agree on the symmetric key.

The same key k appears on both sides, so it must be shared secretly beforehand — the classic "key distribution" problem that Diffie–Hellman later solved.

★ The building blocks

• Block ciphers: encrypt fixed-size chunks — AES (the worldwide standard) works on 128-bit blocks with 128- or 256-bit keys.
• Stream ciphers: generate a keystream and XOR it with the data, bit by bit.
• Modes of operation: CBC, CTR and GCM turn a block cipher into something safe for long messages — GCM even detects tampering.
• Confusion & diffusion: Shannon's two design goals — hide the link between key and ciphertext, and spread each input bit across the whole output.

🔒 The one-time pad: perfect but impractical

XOR your message with a truly random key that is as long as the message and never reused, and the result is provably unbreakable — Shannon's "perfect secrecy." The catch is exactly that key: distributing one-time pads as long as all your data is why we use AES instead.

? What is a protocol?

A single cipher only scrambles data. A protocol is the choreography that combines several tools — key exchange, authentication and encryption — into a conversation that is safe even when strangers are listening. Every HTTPS padlock is one protocol, TLS, running in the background.

A typical secure session does three things: agree on a shared key (key exchange), prove who you're talking to (authentication), then encrypt the data (symmetric crypto).

🤝 The core moves

• Key exchange: Diffie–Hellman / ECDHE let two parties agree a secret over a public wire.
• Message authentication (MAC / HMAC): a keyed tag that proves a message wasn't altered and came from someone who holds the key.
• Digital signatures & certificates: prove identity; a Certificate Authority signs a public key, forming the PKI that browsers trust.
• Forward secrecy: using fresh, throwaway keys each session so a future key leak can't unlock past traffic.

🕵 Proving without revealing

Advanced protocols do surprising things. A zero-knowledge proof convinces someone a statement is true while revealing nothing else — not even the secret behind it. A commitment scheme lets you lock in a value now and reveal it later without being able to change it (it is both binding and hiding), the digital version of a sealed envelope.

? "Looks scrambled" is not enough

Real cryptography never says "it looks random, so it's safe." Instead we define a precise attacker and a precise goal, then prove the attacker can't win. This is what separates modern cryptography from historical code-making.

A scheme is "broken" the moment an attacker does noticeably better than a coin flip — even slightly.

★ The vocabulary of security

• Kerckhoffs's principle: security must rest on the secret key alone — assume the enemy already knows your algorithm.
• Threat models: ciphertext-only → known-plaintext → chosen-plaintext (CPA) → chosen-ciphertext (CCA), each giving the attacker more power.
• Semantic security / IND-CPA: the ciphertext leaks essentially nothing about the plaintext.
• Perfect secrecy: Shannon's gold standard, achieved only by the one-time pad.

🛡 Provable security

The strongest guarantee is a reduction: a proof that if someone could break the scheme, they could also solve a problem everyone believes is hard — like factoring huge numbers. So breaking the cipher is at least as hard as that famous open problem. Security stops being a hope and becomes a theorem.

? Why break your own cipher?

Cryptanalysis is the attacking half of cryptology — and it is how we learn which ciphers to trust. A cipher earns confidence only by surviving decades of the world's best cryptanalysts trying to break it. The attacks below also define the "threat models" that security definitions promise to resist.

★ A toolbox of attacks

• Frequency analysis: count how often each letter appears — this instantly breaks simple substitution ciphers (in English, "E" is most common).
• Brute force: try every key. Feasible against short keys, hopeless against a 128-bit one.
• Differential & linear cryptanalysis: track how tiny input differences ripple through a block cipher — the benchmark every modern cipher must resist.
• Meet-in-the-middle: why Double-DES is barely stronger than DES, so we use Triple-DES instead.
• Side-channel attacks: ignore the math and measure the device — its timing, power draw or emissions — to leak the key.

⌛ The birthday bound

Because of the "birthday paradox," finding two inputs with the same hash takes roughly the square root of the naive effort. That is exactly why secure hashes need 256 bits: it drops the attacker's real work to a still-impossible 2¹²⁸.