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As already discussed by @fgrieu in his answer and myself in the comments of your question and his answer, the standard notion of security of digital signature schemes, namely (strong) existential unforgeability under adaptively chosen message attacks (UF-CMA), does not cover the case you are concerned about. At least for hash-then-sign signatures built ...

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Security properties of hash functions are generally concerned with collision resistance, but preimage resistance is also important. For most common hash functions with an $n$-bit digest size, a successful preimage attack has generic $2^n$ maximum complexity, and a successful collision attack has generic $2^{n/2}$ maximum complexity. Most common hash ...

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Hashes should work on any number of input bits (almost all I/O parameters in cryptography are defined in bits). The output size for collision resistance should be twice the security level. So for a 128 bit level, use 256 bit hashes.

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Reductionist security In a reductionist security proof for some cryptographic protocol $\Pi$ to some alleged hard problem $P$ means, that we can build an algorithm $\cal B$ for solving $P$ if we have access to a hypothetical algorithm $\cal A$ that efficiently breaks the security definition for the protocol $\Pi$. In general, showing a polynomial time ...

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If the salt value is not secret and may be generated at random and stored with the password hash, a large salt value prevents precomputation attacks, including rainbow tables, by ensuring that each user's password is hashed uniquely. This means that two users with the same password will have different password hashes (assuming different salts are used). In ...

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The standard definition of existential forgery allows the adversary to ask and obtain the signature of any message she wants, and claim success if she can exhibit (with sizable odds) any acceptable (message, signature) pair, for any message for which she did not ask signature. Update: There is also strong existential unforgeability, where the adversary ...

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The resume of that other answer could be: When you have a password hashed, it's hard (very hard) to find out what was the original password: you have to try all combinations, until you find the hash. That's brute-force. Someone can speed up a bit this process, by pre-computing many passwords: he'll store all those passwords / hashes, and will try to find ...

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