Abstract
We introduce in this paper cryptographic protocols which use combinatorial group theory. Based on a combinatorial distribution of shares we present secret sharing schemes and cryptosystems using Nielsen transformations. Nielsen transformations are a linear technique to study free groups and general infinite groups. In addition the group of all automorphisms of a free group F, denoted by AUT (F), is generated by a regular Nielsen transformation between two basis of F, and each regular Nielsen transformation between two basis of F defines an automorphism of F.
Highlights
This paper is located in the area of group based cryptography
We introduce in this paper cryptographic protocols which use combinatorial group theory
The free group G is isomorphic to the free group H if and only if we introduce a linear congruence generator because it is used for the cryptosystems in this paper
Summary
This paper is located in the area of group based cryptography. A cryptographic protocol consists of the collection of rules, formulas and methods to handle a cryptographic task. Fine et al 64 protocols, based on non-commutative cryptographic platforms Up to this point the main sources for non-commutative platforms have been nonabelian groups. Important along the line of cryptographic protocols are secret sharing protocols These consist of methods to distribute a secret among a group of users by giving a share of the secret to each. Zhang [3] contains a wealth of information on secret sharing schemes in general and managing an access control group. After introductory definitions we start with a secret sharing scheme using directly the combinatorial distribution of shares. Based on this we present two schemes in which we apply regular Nielsen transformations in connections with faithful representations of free groups and the Nielsen reduction theory. Parts of this paper are from [5]
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