Cryptography is about designing crypto algorithms; cryptanalysis is about breaking them.

Both fall under the umbrella of cryptology, which is the overarching science.

The word "cryptography" comes from the Greek words kryptos (hidden) and graphein (to write) — literally "secret writing."

Tip: Think of it like offense vs. defense — cryptography builds the lock, cryptanalysis tries to pick it.

Go deeper:

• Cryptanalysis - the code-breaking counterpart to cryptography; the two halves of the cryptology umbrella.

Cryptography divides into symmetric (one shared secret key), asymmetric (a public/private key pair), and protocols that combine those primitives — and a complete key-usage split adds the unkeyed primitives, above all hash functions, which use no key at all.

* Cryptography by key usage: symmetric, asymmetric, unkeyed (hash functions), and the protocols built from them. *

Protocols here means cryptographic protocols — multi-step interactions that combine the primitives above to reach a goal no single cipher achieves on its own. Examples: key exchange (Diffie-Hellman), the TLS handshake, authentication schemes, and zero-knowledge proofs.

Where do hash functions go? They take no key, so by key usage they form a class of their own: unkeyed primitives (the classic textbook split is unkeyed / symmetric-key / public-key). Many overviews still fold them into the two keyed branches, because that is where they do their work — as the engine of a MAC such as HMAC (symmetric) and as the digest step of every digital signature (asymmetric). Either way: no key in, fixed-size fingerprint out.

For RSA, the factoring problem is more precisely described as: "if you can factor $N$ into primes, then you can compute the $e$-th root $\bmod N$" — but the reverse has not been proven.

Go deeper:

• Public Key Cryptography - Computerphile - why asymmetric keys exist and how they differ from a shared secret.
• Public-key cryptography - the asymmetric branch of the taxonomy.
• Symmetric-key algorithm - the same-key branch (block and stream ciphers).
• Cryptographic hash function - the unkeyed branch: no key in, fixed-size digest out.