Multiparty Non-Interactive Key Exchange and More from Isogenies on Elliptic Curves

Dan Boneh, Darren Glass, Daniel Krashen, Kristin Lauter, Shahed Sharif, Alice Silverberg, Mehdi Tibouchi, Mark Zhandry

Research output: Contribution to journalArticle

Abstract

We describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n ≥ 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety. Our framework builds a cryptographic invariant map, which is a new primitive closely related to a cryptographic multilinear map, but whose range does not necessarily have a group structure. Nevertheless, we show that a cryptographic invariant map can be used to build several cryptographic primitives, including NIKE, that were previously constructed from multilinear maps and indistinguishability obfuscation.

Original languageEnglish (US)
Pages (from-to)5-14
Number of pages10
JournalJournal of Mathematical Cryptology
Volume14
Issue number1
DOIs
StatePublished - Jan 1 2020

All Science Journal Classification (ASJC) codes

  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics

Keywords

  • Isogenies
  • Multilinear maps
  • Non-Interactive Key Exchange

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    Boneh, D., Glass, D., Krashen, D., Lauter, K., Sharif, S., Silverberg, A., Tibouchi, M., & Zhandry, M. (2020). Multiparty Non-Interactive Key Exchange and More from Isogenies on Elliptic Curves. Journal of Mathematical Cryptology, 14(1), 5-14. https://doi.org/10.1515/jmc-2015-0047