
Previous Article
Efficient reduction of large divisors on hyperelliptic curves
 AMC Home
 This Issue
 Next Article
Relations between arithmetic geometry and public key cryptography
1.  Institute for Experimental Mathematics, University of DuisburgEssen, Ellernstrasse 29, 45326 Essen, Germany 
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 LichtenbaumTate pairing.
[1] 
Anton Stolbunov. Constructing publickey cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215235. doi: 10.3934/amc.2010.4.215 
[2] 
Josep M. Olm, Xavier RosOton. Approximate tracking of periodic references in a class of bilinear systems via stable inversion. Discrete & Continuous Dynamical Systems  B, 2011, 15 (1) : 197215. 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) : 489507. doi: 10.3934/amc.2007.1.489 
[4] 
Felipe Cabarcas, Daniel Cabarcas, John Baena. Efficient publickey operation in multivariate schemes. Advances in Mathematics of Communications, 2019, 13 (2) : 343371. doi: 10.3934/amc.2019023 
[5] 
JoanJosep Climent, Juan Antonio LópezRamos. Public key protocols over the ring $E_{p}^{(m)}$. Advances in Mathematics of Communications, 2016, 10 (4) : 861870. 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) : 13051315. 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) : 221232. 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 hostparasite systems. Global analysis. Discrete & Continuous Dynamical Systems  B, 2007, 8 (1) : 117. 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) : 13991417. doi: 10.3934/mbe.2013.10.1399 
[10] 
Marcy Barge. Pure discrete spectrum for a class of onedimensional substitution tiling systems. Discrete & Continuous Dynamical Systems  A, 2016, 36 (3) : 11591173. doi: 10.3934/dcds.2016.36.1159 
[11] 
Mathias Staudigl, JanHenrik Steg. On repeated games with imperfect public monitoring: From discrete to continuous time. Journal of Dynamics & Games, 2017, 4 (1) : 123. doi: 10.3934/jdg.2017001 
[12] 
Steven D. Galbraith, Ping Wang, Fangguo Zhang. Computing elliptic curve discrete logarithms with improved babystep giantstep algorithm. Advances in Mathematics of Communications, 2017, 11 (3) : 453469. doi: 10.3934/amc.2017038 
[13] 
Gora Adj, Isaac CanalesMartínez, Nareli CruzCortés, Alfred Menezes, Thomaz Oliveira, Luis RiveraZamarripa, Francisco RodríguezHenríquez. Computing discrete logarithms in cryptographicallyinteresting characteristicthree finite fields. Advances in Mathematics of Communications, 2018, 12 (4) : 741759. 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) : 13551374. doi: 10.3934/dcds.2014.34.1355 
[15] 
Andrii Mironchenko, Hiroshi Ito. Characterizations of integral inputtostate stability for bilinear systems in infinite dimensions. Mathematical Control & Related Fields, 2016, 6 (3) : 447466. doi: 10.3934/mcrf.2016011 
[16] 
Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing, Olga Rossi. Invariant nonholonomic Riemannian structures on threedimensional Lie groups. Journal of Geometric Mechanics, 2016, 8 (2) : 139167. 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) : 119. doi: 10.3934/jcd.2017001 
[18] 
Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for nonsymmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems  B, 2009, 11 (3) : 785803. doi: 10.3934/dcdsb.2009.11.785 
[19] 
Susanna Terracini, Juncheng Wei. DCDSA Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems  A, 2014, 34 (6) : iii. doi: 10.3934/dcds.2014.34.6i 
[20] 
Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reactiondiffusionadvection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 23572376. doi: 10.3934/cpaa.2017116 
2017 Impact Factor: 0.564
Tools
Metrics
Other articles
by authors
[Back to Top]