Single-invocation PRF construction from a PRG

Determine whether a pseudorandom function can be constructed using only one invocation of a pseudorandom generator in a black-box construction.

Background

The paper studies lower bounds on fully black-box constructions of pseudorandom functions (PRFs) from pseudorandom generators (PRGs). The GGM construction, combined with domain extension, uses more than logarithmically many PRG invocations, and prior work established optimality only for a restricted class of tree constructions.

The authors explicitly identify the existence of a PRF construction using a single PRG invocation as an unresolved basic challenge. Their results rule out sublinear numbers of calls under particular restrictions, but do not settle the unrestricted single-invocation question.

References

However, the basic challenge of whether a PRF can be built with just one invocation of the PRG still remains open.

— Lower Bounds on Black-Box Constructions of Pseudorandom Functions  (2608.14501 - Alon et al., 14 Aug 2026) in Abstract; Section 1, Introduction

It is an intriguing question whether one could extend the lower bound to all black-box reductions.

— Lower Bounds on Black-Box Constructions of Pseudorandom Functions  (2608.14501 - Alon et al., 14 Aug 2026) in Section 1, Introduction, paragraph “Lower bounds for black-box constructions with non-adaptive calls”