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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus


CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2017: 0.32

Online
ISSN
1869-6104
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Diophantine cryptography in free metabelian groups: Theoretical base

Alexei Myasnikov / Vitalii Roman'kov
  • Omsk State Technical University, Mira, 11, Omsk 644050, Russia; and Institute of Mathematics and Information Technologies, Omsk F. M. Dostoevsky State University, Mira, 55-A, Omsk 644077, Russia
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Published Online: 2014-10-15 | DOI: https://doi.org/10.1515/gcc-2014-0011

Abstract

In this paper we study so-called Diophantine cryptology, a collection of cryptographic schemes where the computational security assumptions are based on hardness of solving some Diophantine equations, and some general ideas and techniques that occur in this area. In particular, we study an interesting variation of the endomorphism problem in groups, termed the double endomorphism problem. We prove that this problem is undecidable in free metabelian groups of sufficiently large rank. We relate this result to computational security assumptions of some group-based cryptosystems. In particular, we show how to improve the Grigoriev–Shpilrain's protocol to get a new computational security assumption based on the double endomorphism problem, providing a better theoretical foundation to security.

Keywords: Free metabelian group; endomorphism problem; cryptosystems; authentication

MSC: 20F10; 68W30; 20F16; 11T71

About the article

Received: 2014-09-15

Published Online: 2014-10-15

Published in Print: 2014-11-01


Funding Source: Russian Science Foundation

Award identifier / Grant number: 14-11-00085

Funding Source: NSF

Award identifier / Grant number: DMS-1318716

Funding Source: NSF

Award identifier / Grant number: DMS-1201550

Funding Source: NSA

Award identifier / Grant number: H98230-14-1-0128


Citation Information: Groups Complexity Cryptology, Volume 6, Issue 2, Pages 103–120, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2014-0011.

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