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
[1]

Gerhard Frey. Relations between arithmetic geometry and public key cryptography. Advances in Mathematics of Communications, 2010, 4 (2) : 281-305. doi: 10.3934/amc.2010.4.281

[2]

Yuri B. Gaididei, Rainer Berkemer, Carlos Gorria, Peter L. Christiansen, Atsushi Kawamoto, Takahiro Shiga, Mads P. Sørensen, Jens Starke. Complex spatiotemporal behavior in a chain of one-way nonlinearly coupled elements. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1167-1179. doi: 10.3934/dcdss.2011.4.1167

[3]

Giacomo Micheli. Cryptanalysis of a noncommutative key exchange protocol. Advances in Mathematics of Communications, 2015, 9 (2) : 247-253. doi: 10.3934/amc.2015.9.247

[4]

Mohammad Sadeq Dousti, Rasool Jalili. FORSAKES: A forward-secure authenticated key exchange protocol based on symmetric key-evolving schemes. Advances in Mathematics of Communications, 2015, 9 (4) : 471-514. doi: 10.3934/amc.2015.9.471

[5]

Joan-Josep Climent, Juan Antonio López-Ramos. Public key protocols over the ring $E_{p}^{(m)}$. Advances in Mathematics of Communications, 2016, 10 (4) : 861-870. doi: 10.3934/amc.2016046

[6]

Bin Chen, Xiongping Dai. On uniformly recurrent motions of topological semigroup actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2931-2944. doi: 10.3934/dcds.2016.36.2931

[7]

Rainer Steinwandt, Adriana Suárez Corona. Cryptanalysis of a 2-party key establishment based on a semigroup action problem. Advances in Mathematics of Communications, 2011, 5 (1) : 87-92. doi: 10.3934/amc.2011.5.87

[8]

Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215

[9]

Lixin Xu, Wanquan Liu. A new recurrent neural network adaptive approach for host-gate way rate control protocol within intranets using ATM ABR service. Journal of Industrial & Management Optimization, 2005, 1 (3) : 389-404. doi: 10.3934/jimo.2005.1.389

[10]

Roland Martin. On simple Igusa local zeta functions. Electronic Research Announcements, 1995, 1: 108-111.

[11]

Thierry Barbot, Carlos Maquera. On integrable codimension one Anosov actions of $\RR^k$. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 803-822. doi: 10.3934/dcds.2011.29.803

[12]

Danijela Damjanović. Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions. Journal of Modern Dynamics, 2007, 1 (4) : 665-688. doi: 10.3934/jmd.2007.1.665

[13]

Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985

[14]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

[15]

Flávia M. Branco. Sub-actions and maximizing measures for one-dimensional transformations with a critical point. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 271-280. doi: 10.3934/dcds.2007.17.271

[16]

Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191

[17]

Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753

[18]

Tsuyoshi Kajiwara, Toru Sasaki, Yasuhiro Takeuchi. Construction of Lyapunov functions for some models of infectious diseases in vivo: From simple models to complex models. Mathematical Biosciences & Engineering, 2015, 12 (1) : 117-133. doi: 10.3934/mbe.2015.12.117

[19]

Florian Luca, Igor E. Shparlinski. On finite fields for pairing based cryptography. Advances in Mathematics of Communications, 2007, 1 (3) : 281-286. doi: 10.3934/amc.2007.1.281

[20]

Christoph Hauert, Nina Haiden, Karl Sigmund. The dynamics of public goods. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 575-587. doi: 10.3934/dcdsb.2004.4.575

2017 Impact Factor: 0.564

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]