**On the Inner Product Predicate and a Generalization of Matching Vector Families**

*Balthazar Bauer and Jevgēnijs Vihrovs and Hoeteck Wee*

**Abstract: **Motivated by cryptographic applications such as predicate encryption, we consider the problem of representing an arbitrary predicate as the inner product predicate on two vectors. Concretely, fix a Boolean function $P$ and some modulus $q$. We are interested in encoding $x$ to $\vec x$ and $y$ to $\vec y$ so that $$P(x,y) = 1 \Longleftrightarrow \langle\vec x,\vec y\rangle= 0 \bmod q,$$ where the vectors should be as short as possible. This problem can also be viewed as a generalization of matching vector families, which corresponds to the equality predicate. Matching vector families have been used in the constructions of Ramsey graphs, private information retrieval (PIR) protocols, and more recently, secret sharing.

Our main result is a simple lower bound that allows us to show that known encodings for many predicates considered in the cryptographic literature such as greater than and threshold are essentially optimal for prime modulus $q$. Using this approach, we also prove lower bounds on encodings for composite $q$, and then show tight upper bounds for such predicates as greater than, index and disjointness.

**Category / Keywords: **Predicate Encryption, Inner Product Encoding, Matching Vector Families

**Original Publication**** (with minor differences): **FSTTCS 2018

**Date: **received 4 Oct 2018, last revised 4 Oct 2018

**Contact author: **jevgenijs vihrovs at lu lv

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

**Version: **20181009:155042 (All versions of this report)

**Short URL: **ia.cr/2018/945

[ Cryptology ePrint archive ]