Which of these statements are true or false: "A cryptographically secure hash function is also a one-way function" and "A one-way function is also a cryptographically secure hash function"?

A crypto-secure hash function IS a one-way function (true). But a one-way function is NOT necessarily a crypto-secure hash function (false).

A useful slogan: "A cryptographically secure hash function $\Rightarrow$ is a one-way function." The reverse arrow does NOT hold.

Why hash → one-way? A crypto-secure hash function has preimage resistance — given $h(x)$, you can't find $x$. That's exactly the definition of one-way.

Why one-way ↛ hash?

• A one-way function doesn't need to have a fixed-size output
• A one-way function doesn't need collision resistance
• Example: RSA encryption $y = x^e \mod N$ is a one-way function (with trapdoor), but it's not a hash function
• Example: A checksum is a hash function but NOT one-way (not cryptographically secure)

Related: One-way functions are a broader category that includes:

• Hash functions (without key, like SHA-3)
• Keyed hash functions (MAC, HMAC)
• Key derivation functions (KDF)
• Functions underlying asymmetric crypto (factoring, discrete log, ECC)

Go deeper:

• One-way function (Wikipedia) — the broad class and how a crypto hash is one particular (fixed-output, collision-resistant) instance.