# American Institute of Mathematical Sciences

November  2007, 1(4): 489-507. doi: 10.3934/amc.2007.1.489

## Public key cryptography based on semigroup actions

 1 Department of Mathematics, University of Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland 2 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States 3 Institut für Mathematik, Universität Zürich, Zürich, CH-8057

Received  June 2007 Revised  October 2007 Published  October 2007

A generalization of the original Diffie-Hellman key exchange in $(\mathbb Z$∕$p\mathbb Z)$* found a new depth when Miller [27] and Koblitz [16] suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general.
In Section 5 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.
Citation: Gérard Maze, Chris Monico, Joachim Rosenthal. Public key cryptography based on semigroup actions. Advances in Mathematics of Communications, 2007, 1 (4) : 489-507. doi: 10.3934/amc.2007.1.489
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