Pseudorandom Function (PRF)

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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 functions , with respect to a keyspace , domain , and range , such that:

  • , takes a security parameter, and outputs a key ,
  • , takes a key and input , and outputs an element .

Generally, we assume that \$|\mathcal{D}|\$ 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 only different in this definition is that we sample $s(\lambda)$ 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.


Relationship to other primitives

There are also many folklore results surrounding pseudorandom functions:

  • PRPs over are PRFs, due to the switching lemma [[[folklore]]]


Sufficient assumptions


Variations and Related Primitives