Semigroup Forum (2010) 81: 172–186 DOI 10.1007/s00233-010-9248-8 RESEARCH ARTICLE Properties of certain semigroups and their potential as platforms for cryptosystems P.H. Kropholler · S.J. Pride · W.A.M. Othman · K.B. Wong · P.C. Wong Dedicated to the memory of Walter Douglas Munn. Received: 30 December 2009 / Accepted: 15 April 2010 / Published online: 16 July 2010 © Springer Science+Business Media, LLC 2010 Abstract In this paper, we study some properties of semigroups with presentation a,b ; a p = b r ,a q = b s . We will also study their potential as platforms for the Diffie-Hellman key exchange protocol. Keywords Semigroups · Diffie-Hellman · Discrete logarithm problem · Key exchange protocol · Cryptosystems 1 Introduction In the last ten or more years, there has been a lot of on-going research work to find more secure and efficient public key cryptosystems based on objects like non-abelian Funded by the British Council PMI-2-RC150 contract, “Group Theory and Cryptography”, S.J. Pride and P.C. Wong, senior investigators. P.H. Kropholler · S.J. Pride ( ) Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotland, UK e-mail: stephen.pride@glasgow.ac.uk P.H. Kropholler e-mail: peter.kropholler@glasgow.ac.uk W.A.M. Othman · K.B. Wong · P.C. Wong Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia W.A.M. Othman e-mail: wanainun@um.edu.my K.B. Wong e-mail: kbwong@um.edu.my P.C. Wong e-mail: wongpc@um.edu.my