Disclaimer: I'm new to cryptography.
Background: From Applied Cryptography (Bruce Schneier), page 30 of 2nd edition
A trapdoor one-way function is a special type of one-way function, one with a secret trapdoor. It is easy to compute in one direction and hard to compute in the other direction. But, if you know the secret, you can easily compute the function in the other direction. That is, it is easy to compute $f(x)$ given $x$, and hard to compute $x$ given $f(x)$. However, there is some secret information, $y$, such that given $f(x)$ and $y$ it is easy to compute $x$.
Question: From a high level (i.e., more formally), is the following correct?
m = message i = input (e.g., public key) t = trapdoor (e.g., private key) h = one-way hash f(m,i) = h f(h,t) = m
Edit: Please see Schneier's use of a "one-way hash" on page 38 of the same reference:
(..) Alice signs the hash of the document. In this protocol, both the one-way hash function and the digital signature algorithm are agreed upon beforehand.
- Alice produces a one-way hash of a document.
- Alice encrypts the hash with her private key, thereby signing the document.
- Alice sends the document and the signed hash to Bob.
- Bob produces a one-way hash of the document that Alice sent. He then, using the digital signature algorithm, decrypts the signed hash with Alice’s public key. If the signed hash matches the hash he generated, the signature is valid.
Edit 2: Update, based on suggested answers:
x = message i = input (e.g., public key) t = trapdoor (e.g., private key) h = one-way hash f(x) = h g(h,t) = x