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

TL;DR

This paper studies how to encode arbitrary predicates as inner product relations on vectors, providing lower bounds and optimal constructions for various cryptographic predicates, extending the concept of matching vector families.

Contribution

It introduces a simple lower bound framework for encoding predicates as inner products, establishing optimal bounds for many cryptographic predicates over prime and composite moduli.

Findings

Lower bounds for predicate encodings over prime moduli.

Tight upper bounds for predicates like greater than, index, and disjointness.

Optimal encoding sizes for various cryptographic predicates.

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.