UOWHFs are proposed as an alternative to collision-resistant hash functions (CRHFs).
In contrast, UOWHFs require that it be hard to find a collision where one preimage is chosen independently of the hash function parameters.
The primitive was suggested by Moni Naor and Moti Yung and is also known as "target collision resistant" hash functions; it was employed to construct general digital signature schemes without trapdoor functions, and also within chosen-ciphertext secure public key encryption schemes.
The UOWHF family contains a finite number of hash functions with each having the same probability of being used.
UOWHFs are thought to be less computationally expensive than CRHFs, and are most often used for efficiency purposes in schemes where the choice of the hash function happens at some stage of execution, rather than beforehand.