Pseudorandom Function (PRF): Difference between revisions

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=== Syntax ===
=== Syntax ===
A Pseudorandom Function (PRF) is a tuple of functions <math>(\mathsf{Gen}, F)</math>, with respect to a keyspace <math>\mathcal{K}</math>, domain <math>\mathcal{D}</math>, and range <math>\mathcal{R}</math>, such that:
A Pseudorandom Function (PRF) is a tuple of functions <math>(\mathsf{Gen}, F)</math>, with respect to a keyspace <math>\mathcal{K}</math>, domain <math>\mathcal{D}</math>, and range <math>\mathcal{R}</math>, such that:
* <math>\mathsf{Gen}(1^{\lambda}) \to k</math>, takes a security parameter, and outputs a key <math>k\in \mathcal{K}</math>,
* <math>\mathsf{Gen}(1^{\lambda}) \to k</math>, is a randomized algorithm that takes a security parameter, and outputs a key <math>k\in \mathcal{K}</math>,
* <math>F_k(x) \to y</math>, takes a key <math>k\in \mathcal{K}</math> and input <math>x\in \mathcal{D}</math>, and outputs an element <math>y\in \mathcal{R}</math>.
* <math>F_k(x) \to y</math>, is a deterministic algorithm that takes a key <math>k\in \mathcal{K}</math> and input <math>x\in \mathcal{D}</math>, and outputs an element <math>y\in \mathcal{R}</math>.


Generally, we assume that <math>|\mathcal{D}|</math> is "large," in the sense that it grows exponentially with the security parameter. If instead <math>|\mathcal{D}|</math> bounded by some polynomial in the security parameter, then the primitive is a "small-domain" PRF.
Generally, we assume that <math>|\mathcal{D}|</math> is "large," in the sense that it grows exponentially with the security parameter. If instead <math>|\mathcal{D}|</math> bounded by some polynomial in the security parameter, then the primitive is a "small-domain" PRF.

Revision as of 13:52, 4 July 2024


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:

  • , 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 $s(\lambda)$ 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 in the stronger definition.

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

Other Notes