Aggregate Pseudorandom Functions and Connections to Learning

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. Aggregate pseudo-random functions fall within the framework of the work of Goldreich, Goldwasser, and Nussboim [GGN10] on the “Implementation of Huge Random Objects,” providing truthful implementations of pseudorandom functions for which aggregate queries can be answered. 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. ∗MIT. †MIT and the Weizmann Institute of Science. ‡MIT. 1 c ©IACR 2015. This article is a minor revision of the version published by Springer-Verlag. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 9 (2015)