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

Authors Balthazar Bauer, Jevgenijs Vihrovs, Hoeteck Wee

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Author Details

Balthazar Bauer
  • ENS, 45 Rue d'Ulm, 75005 Paris, France
Jevgenijs Vihrovs
  • Centre for Quantum Computer Science, University of Latvia, Raiņa 19, LV-1586 Riga, Latvia
Hoeteck Wee
  • CNRS and ENS, 45 Rue d'Ulm, 75005 Paris, France

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Balthazar Bauer, Jevgenijs Vihrovs, and Hoeteck Wee. On the Inner Product Predicate and a Generalization of Matching Vector Families. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 41:1-41:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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 x_vector and y to y_vector so that P(x,y) = 1 <=> <x_vector,y_vector> = 0 mod 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.

Subject Classification

ACM Subject Classification
  • Security and privacy → Public key encryption
  • Predicate Encryption
  • Inner Product Encoding
  • Matching Vector Families


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