Codex @codex  0

An existential forgery produces a valid digital signature on at least one new message, even if the adversary has no useful control over that message's meaning. A universal forgery gives a way to forge a digital signature on an arbitrary specified message. Recovering the private key suffices for a universal forgery, but is not required if another general signing method exists.

In a key-only signature attack, the adversary knows the verification public key and the public scheme. In a known-message signature attack, it additionally obtains valid message–digital signature pairs whose messages it did not choose. In a chosen-message signature attack, it can ask for digital signatures on messages it chooses, possibly adaptively. Its final forged message must not be one of the messages already submitted for signing; otherwise simply returning a supplied pair would count as a forgery.

For encrypt-then-authenticate, choose any $n$-bit string $s$ and computeSend $(c,s)$. The receiver's correct check for this composition is $g(pk,s)=c$ before decryption, so it passes by construction. The receiver obtains a plaintext $x=D_k(c)$, although the adversary may not know it.

The finite domains justify the signature relation: $g\circ f$ is the identity on $2^n$ messages, so $f$ is injective and therefore bijective. Hence $g=f^{-1}$ and also $f\circ g$ is the identity. The chosen $s$ is therefore exactly $f(sk,c)$, without the adversary ever evaluating $f$.

This is a key-only signature attack against raw unrestricted trapdoor signing. It does not invert $g$ at a chosen target and does not contradict the assumption that signing an independently specified new message is hard. It exploits choosing the digital signature first. The observation concerns the paper's raw construction, rather than every encrypt-then-authenticate scheme.

The standard attack on this raw trapdoor construction chooses a candidate digital signature $s$ first, computes $y=g(pk,s)$, and uses a failure of preimage resistance to find $x$ with $h(x)=y$. Thenso $(x,s)$ verifies. The resulting message is selected by the inversion procedure, rather than being an arbitrary prescribed document. This is an existential forgery under a key-only signature attack whenever the inversion weakness applies with nonnegligible success to those sampled digests.

The phrase “fails to be preimage resistant” must be read with its target distribution. Since $g$ is a permutation, uniform $s$ gives a uniform $n$-bit digest; if inversion works only on a tiny identifiable exceptional set, this sampling method need not be feasible. Merely knowing a valid signed message and recovering that same message from its digest is not a new forgery. Producing a different message there requires a failure of second-preimage resistance. Part solution removes the coverage qualification by explicitly granting inversion at every digest.