Alternative security models for a pseudorandom function

In the paper we analyze various security models for pseudorandom functions that arise in the analysis of cryptographic protocols (such as 5G-AKA) and study the reducibility of non-standard pseudorandomness models to the standard $\mathsf{PRF}$ model.
We consider several models. (a) $\mathsf{PRF}^+$ model formalizes the following requirement: the outputs of a pseudorandom function on adaptively selected inputs must be indistinguishable from random binary strings of the appropriate length, even if the adversary has the opportunity to receive as “an additional information” the outputs of a “real” pseudorandom function. (b) $\mathsf{UF}$-$\mathsf{PRF}$ model formalizes the requirement that it is impossible to forge the value of a pseudorandom function on a fresh input (similar to the models for the MAC function). (c) $\mathsf{LOR}$-$\mathsf{PRF}$ model formalizes the indistinguishability property of “cryptographic bindings” calculated via pseudorandom function on different keys.
We also study the natural generalization of these models to the case of multiple users in the system ($\mathsf{mPRF}^+$ and $\mathsf{mUF}$-$\mathsf{PRF}$ models). We show that these new models can be reduced to the basic $\mathsf{PRF}$ model for a pseudorandom function family. The results can be used in the analysis of various cryptographic protocols.