Pseudorandom functions with structure : extensions and implications

Abstract

In the first part of this work, we introduce a new type of pseudo-random function for which "aggregate queries" over exponential-sized sets can be efficiently answered. We show how to use algebraic properties of underlying classical pseudo random functions, to construct such "aggregate pseudo-random functions" for a number of classes of aggregation queries under cryptographic hardness assumptions. For example, one aggregate query we achieve is the product of all function values accepted by a polynomial-sized read-once boolean formula. On the flip side, we show that certain aggregate queries are impossible to support. In the second part of this work, we show how various extensions of pseudo-random functions considered recently in the cryptographic literature, yield impossibility results for various extensions of machine learning models, continuing a line of investigation originated by Valiant and Kearns in the 1980s. The extended pseudo-random functions we address include constrained pseudo random functions, aggregatable pseudo random functions, and pseudo random functions secure under related-key attacks. In the third part of this work, we demonstrate limitations of the recent notions of constrained pseudo-random functions and cryptographic watermarking schemes. Specifically, we construct pseudorandom function families that can be neither punctured nor watermarked. This is achieved by constructing new unobfuscatable pseudorandom function families for new ranges of parameters.