**Big Subset and Small Superset Obfuscation**

*Steven D. Galbraith and Trey Li*

**Abstract: **Let S = {1,...,n} be a set of integers and X be a subset of S. We study the boolean function f_X(Y) which outputs 1 if and only if Y is a big enough subset (resp. small enough superset) of X. Our purpose is to protect X from being known, yet allow evaluations of f_X on any input Y \subseteq S. The corresponding research area is called function obfuscation. The two kinds of functions are called big subset functions (BSF) and small superset functions (SSF) respectively.

In this paper, we obfuscate BSF and SSF in a very simple and efficient way. We prove both virtual black-box (VBB) security and input-hiding security in the standard model based on the subset product problem.

We also give a proof of input-hiding based on the discrete logarithm problem (DLP) for the conjunction obfuscation by Bartusek et al. [5] (see Appendix A) and propose a new conjunction obfuscation based on BSF and SSF obfuscation (see Appendix B). The security of our conjunction obfuscation is from our new computational problem called the Twin Subset Product Problem.

**Category / Keywords: **public-key cryptography / Obfuscation, Big Subset, Small Superset, Conjunction, Twin Subset Product Problem

**Date: **received 23 Aug 2020, last revised 13 Nov 2020

**Contact author: **s galbraith at auckland ac nz,trey li@auckland ac nz

**Available format(s): **PDF | BibTeX Citation

**Version: **20201114:010332 (All versions of this report)

**Short URL: **ia.cr/2020/1018

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