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

Author:

Boneh Dan1,Glass Darren2,Krashen Daniel3,Lauter Kristin4,Sharif Shahed5,Silverberg Alice6,Tibouchi Mehdi7,Zhandry Mark8

Affiliation:

1. Stanford University, California, United States of America

2. Gettysburg College, Gettysburg, United States of America

3. Rutgers University, New Jersey, United States of America

4. Microsoft Research, Bellevue, United States of America

5. California State University San Marcos, California, United States of America

6. University of California, Irvine, United States of America

7. NTT Corporation, Tokyo, Japan

8. Princeton University, Princeton, United States of America

Abstract

AbstractWe 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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Computational Mathematics,Computer Science Applications

Reference84 articles.

1. Applications of multilinear forms to cryptography;Contemporary Mathematics,2003

2. A class number associated with a product of two elliptic curves;Natur. Sci. Rep. Ochanomizu Univ.,1965

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