A one-way function is easy to compute and practically impossible to undo, like scrambling an egg. Hashes, encryption and signatures all build on this bet.
The idea: Almost every tool on the map has a one-way function as its engine. Store only the hash of a password, and a thief who steals it can't run it backwards. Feed a secret key and a counter into a one-way-style function and you get an endless stream that looks random: that's a cipher. Even signatures can be built from nothing else: publish the hashes of some secret values, and later sign by revealing some of the secrets. What one-way functions can't obviously do is let two strangers agree on a secret in public. That needs the extra structure of the public-key box.
Why it matters: It's the bedrock under everything on this map.
On the map: the minimal assumption behind all computational crypto: hashes, symmetric crypto, signatures. Key exchange needs more (KE node)
A one-way function is easy to compute and infeasible to reverse. It is the minimal assumption behind all computational cryptography: hashes like Keccak and SHA-256, symmetric encryption, and signatures can all be built from one-way functions[1, 2].
Key exchange, the thing that lets two Ethereum nodes agree on a shared key over the internet, needs more than that[3]. See the key-exchange node. Hash-based schemes, which the post-quantum roadmap leans on, need only this kind of assumption.
If any of these breaks, this breaks too.
If this breaks, these break with it.