Hash functions are irreversible; a potentially-infinite number of inputs all map to the same output.
I think the one-way functions referred to are what used to be called trap-door functions. They're not irreversible, like hash functions; they're computationally hard to reverse, unless you happen to know the key to open the trap-door.
Even if you can’t get back to the original data you hashed, you’d still only need to pick an input length and then compute a single example that produced that hash to break all the cryptographic uses of hash functions like signatures and password storage though.
>Hash functions are irreversible; a potentially-infinite number of inputs all map to the same output.
I don't think that's pre-image resistance, or else extremely trivial "Hash functions" like
- Line up the input string into a matrix of n columns, XOR every column into a single bit, output the resulting n-bit hash.
Would be pre-image resistant, but it's obviously not. Given an n-bit hash I can trivially generate a message that hashes to it.
Generally, there are at least 3 notions of irreversiblity that might get confused:
- Non-bijectivity : That just means the set of images is smaller than the set of pre-images, so it's impossible (in general) to tell what input produced a given image. If F(x) == 1 for all x in X, it's impossible to tell which x was evaluated in order to get a 1. This is what you described, and it's - by itself - useless security-wise.
- One-way functions : Given x, its reasonably efficient to calculate y = F(x). But given y = F(x), it's prohibitively expensive to recover x = F^-1(y). The function is easy to compute in one direction, and practically impossible in the other. The inverse might very will exist, maybe even infinite inverses exist for every y, but you would need the computational resources of a thousand thousand universe working for a thousand thousand epoch to even get close to finding a single one.
- Trap-door functions : They are special one-way functions that have the property that : given a special piece of information about x or y (an inequality, a prime factor,....), x = F^-1(y) suddenly becomes much easier to compute, maybe even as easy as y = F(x).
One-way functions are definitely not trap-door functions, they are a weaker assumption. But you can get all symmetric cryptography out of a one-way function; it doesn't have to be reversible.
For instance, you can use a hash function to encrypt, by XORing its output with a plaintext, using it as a stream cipher.
As I mentioned in my reply to the sibling comment, a hash function is enough. For example, you could encrypt by feeding a key + block counter to a hash function and XORing the plaintext blocks with the output. The recipient with the same key can generate the same digests and XOR to decrypt.
That's not a proof, of course, it's just to give an intuition for how all symmetric cryptography can be constructed from a one-way function.
AES would still be totally okay.