American Institute of Mathematical Sciences

Advanced
May  2010, 4(2): 281-305. doi: 10.3934/amc.2010.4.281

Relations between arithmetic geometry and public key cryptography

Gerhard Frey 1,

1. 

Institute for Experimental Mathematics, University of Duisburg-Essen, Ellernstrasse 29, 45326 Essen, Germany

Received  July 2009 Revised  November 2009 Published  May 2010

In the article we shall try to give an overview of the interplay between the design of public key cryptosystems and algorithmic arithmetic geometry. We begin in Section 2 with a very abstract setting and try to avoid all structures which are not necessary for protocols like Diffie-Hellman key exchange, ElGamal signature and pairing based cryptography (e.g. short signatures). As an unavoidable consequence of the generality the result is difficult to read and clumsy. But nevertheless it may be worthwhile because there are suggestions for systems which do not use the full strength of group structures (see Subsection 2.2.1) and it may motivate to look for alternatives to known group-based systems.
  But, of course, the main part of the article deals with the usual realization by discrete logarithms in groups, and the main source for cryptographically useful groups are divisor class groups.
  We describe advances concerning arithmetic in such groups attached to curves over finite fields including addition and point counting which have an immediate application to the construction of cryptosystems.
  For the security of these systems one has to make sure that the computation of the discrete logarithm is hard. We shall see how methods from arithmetic geometry narrow the range of candidates usable for cryptography considerably and leave only carefully chosen curves of genus $1$ and $2$ without flaw.
  A last section gives a short report on background and realization of bilinear structures on divisor class groups induced by duality theory of class field theory, the key concept here is the Lichtenbaum-Tate pairing.
Keywords: bilinear structures, public key systems., Discrete logarithms, divisor class groups.
Mathematics Subject Classification: 11R65, 11R37, 11G2.
Citation: 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
[1]

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
[2]

Josep M. Olm, Xavier Ros-Oton. Approximate tracking of periodic references in a class of bilinear systems via stable inversion. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 197-215. doi: 10.3934/dcdsb.2011.15.197
[3]

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
[4]

Felipe Cabarcas, Daniel Cabarcas, John Baena. Efficient public-key operation in multivariate schemes. Advances in Mathematics of Communications, 2019, 13 (2) : 343-371. doi: 10.3934/amc.2019023
[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]

Frédéric Bernicot, Vjekoslav Kovač. Sobolev norm estimates for a class of bilinear multipliers. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1305-1315. doi: 10.3934/cpaa.2014.13.1305
[7]

El Hassan Zerrik, Nihale El Boukhari. Optimal bounded controls problem for bilinear systems. Evolution Equations & Control Theory, 2015, 4 (2) : 221-232. doi: 10.3934/eect.2015.4.221
[8]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1
[9]

Hui Cao, Yicang Zhou, Zhien Ma. Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1399-1417. doi: 10.3934/mbe.2013.10.1399
[10]

Marcy Barge. Pure discrete spectrum for a class of one-dimensional substitution tiling systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1159-1173. doi: 10.3934/dcds.2016.36.1159
[11]

Mathias Staudigl, Jan-Henrik Steg. On repeated games with imperfect public monitoring: From discrete to continuous time. Journal of Dynamics & Games, 2017, 4 (1) : 1-23. doi: 10.3934/jdg.2017001
[12]

Steven D. Galbraith, Ping Wang, Fangguo Zhang. Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm. Advances in Mathematics of Communications, 2017, 11 (3) : 453-469. doi: 10.3934/amc.2017038
[13]

Gora Adj, Isaac Canales-Martínez, Nareli Cruz-Cortés, Alfred Menezes, Thomaz Oliveira, Luis Rivera-Zamarripa, Francisco Rodríguez-Henríquez. Computing discrete logarithms in cryptographically-interesting characteristic-three finite fields. Advances in Mathematics of Communications, 2018, 12 (4) : 741-759. doi: 10.3934/amc.2018044
[14]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[15]

Andrii Mironchenko, Hiroshi Ito. Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions. Mathematical Control & Related Fields, 2016, 6 (3) : 447-466. doi: 10.3934/mcrf.2016011
[16]

Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing, Olga Rossi. Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. Journal of Geometric Mechanics, 2016, 8 (2) : 139-167. doi: 10.3934/jgm.2016001
[17]

Jake Bouvrie, Boumediene Hamzi. Kernel methods for the approximation of some key quantities of nonlinear systems. Journal of Computational Dynamics, 2017, 4 (1&2) : 1-19. doi: 10.3934/jcd.2017001
[18]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785
[19]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i
[20]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences