Separating the power of monotone span programs over different fields

Amos Beimel, Enav Weinreb

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

Monotone span programs are a linear-algebraic model of computation. They are equivalent to linear secret sharing schemes and have various applications in cryptography and complexity. A fundamental question is how the choice of the field in which the algebraic operations are performed effects the power of the span program. In this paper we prove that the power of monotone span programs over finite fields of different characteristics is incomparable; we show a super-polynomial separation between any two fields with different characteristics, answering an open problem of Pudlák and Sgall (1998). Using this result we prove a super-polynomial lower bound for monotone span programs for a function in uniform - NC2 (and therefore in P), answering an open problem of Babai, Wigderson, and Gál (1999). Finally, we show that quasi-linear schemes, a generalization of linear secret sharing schemes introduced in Beimel and Ishai (2001), are stronger than linear secret sharing schemes. In particular, this proves, without any assumptions, that non-linear secret sharing schemes are more efficient than linear secret sharing schemes.

Original languageEnglish GB
Title of host publicationProceedings - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003
PublisherIEEE Computer Society
Pages428-437
Number of pages10
ISBN (Electronic)0769520405
DOIs
StatePublished - 1 Jan 2003
Event44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 - Cambridge, United States
Duration: 11 Oct 200314 Oct 2003

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2003-January
ISSN (Print)0272-5428

Conference

Conference44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003
Country/TerritoryUnited States
CityCambridge
Period11/10/0314/10/03

Keywords

  • Arithmetic
  • Circuits
  • Combinatorial mathematics
  • Computational complexity
  • Computational modeling
  • Computer science
  • Cryptography
  • Galois fields
  • Linear algebra
  • Vectors

ASJC Scopus subject areas

  • Computer Science (all)

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