Pseudorandom Function (PRF)
A Pseudorandom Function (PRF) is a primitive that allows someone to succinctly represent a function that is indistinguishable from a random function. A user of a PRF generates a key, which it can use to evaluate a function at many points. Any efficient adversary, who only sees these inputs and outputs of the keyed function, cannot distinguish them from a random function.
PRFs were originally defined by [GGM84] and are a basic building block for many more complex primitives such as Symmetric Encryption (SE).
Formal Definition
Syntax
A Pseudorandom Function (PRF) is a tuple of efficient functions , with respect to a keyspace , domain , and range , such that:
- , is a randomized algorithm that takes a security parameter, and outputs a key ,
- , is a deterministic algorithm that takes a key and input , and outputs an element .
Generally, we assume that is "large," in the sense that it grows exponentially with the security parameter. If instead bounded by some polynomial in the security parameter, then the primitive is a "small-domain" PRF.
Security
A PRF is secure if for all efficient , there exists a negligible function , such that
Weak Security
A PRF is weakly secure if for all efficient , and all polynomials , there exists a negligible function , such that
The difference in this definition is that we sample inputs at random instead of allowing a distinguisher to choose them. This is strictly weaker, as a distinguisher could always query their oracle at random locations in the stronger definition.
Relationship to other primitives
- PRGs imply the existence of PRFs, due to [GGM84] (this means that OWFs imply the existence of PRFs, in combination with [HILL99])
- PRFs imply the existence of PRPs, due to [LR88]
There are also many folklore results surrounding pseudorandom functions: