Diophantine cryptography in free metabelian groups: Theoretical base
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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.
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
© 2014 by De Gruyter
Articles in the same Issue
- Frontmatter
- Editorial
- Friends and relatives of BS(1,2)
- Reflections on some aspects of infinite groups
- Generalized small cancellation presentations for automatic groups
- Diophantine cryptography in free metabelian groups: Theoretical base
- Palindromic width of wreath products, metabelian groups, and max-n solvable groups
- Group-theoretic orbit decidability
- Decoy-based information security
Articles in the same Issue
- Frontmatter
- Editorial
- Friends and relatives of BS(1,2)
- Reflections on some aspects of infinite groups
- Generalized small cancellation presentations for automatic groups
- Diophantine cryptography in free metabelian groups: Theoretical base
- Palindromic width of wreath products, metabelian groups, and max-n solvable groups
- Group-theoretic orbit decidability
- Decoy-based information security
