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What's the difference between UOWHF and CRHF and why are UOWHF useful?

As far as I understand, Universal One-Way Hash Functions are an alternative to CRHF. While for CRHF it is hard, given randomly chosen hash function parameters, to find any collision of the hash function; for UOWHF it's hard to find a collision where one preimage is chosen independently of the hash function parameters.

How does this difference affect the behaviour of the functions and why are UOWHF so relevant?

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It's a combination of:

  1. UOWHF is a weaker primitive than CRHF (collision resistance implies universal one-wayness) and it turns out that it can be (black-box) constructed from one-way functions (OWFs) [NY,R]. While, CRHFs cannot be (black-box) constructed from OWFs [S].
  2. For some applications, (e.g., digital signatures [NY]) UOWHFs suffice instead of collision resistance.

[NY]: Naor and Yung, Universal One-Way Hash Functions and their Cryptographic Applications, STOC'89

[R]: Rompel, One-way functions are necessary and sufficent for secure signatures,STOC'90

[S]: Simon. Finding collisions on a one-way street: Can secure hash functions be based on general assumptions?, Eurocrypt'98.

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