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. 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 pseudo-random 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.
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Cohen, A., Goldwasser, S., Vaikuntanathan, V. (2015). Aggregate Pseudorandom Functions and Connections to Learning. In: Dodis, Y., Nielsen, J.B. (eds) Theory of Cryptography. TCC 2015. Lecture Notes in Computer Science, vol 9015. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46497-7_3
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