November  2014, 8(4): 437-458. doi: 10.3934/amc.2014.8.437

The geometry of some parameterizations and encodings

1. 

Université de Bordeaux, IMB, UMR 5251, F-33400 Talence, France

2. 

DGA.MI, La Roche Marguerite, F-35174 Bruz, France

Received  May 2014 Revised  June 2014 Published  November 2014

We explore parameterizations by radicals of low genera algebraic curves. We prove that for $q$ a prime power that is large enough and prime to $6$, a fixed positive proportion of all genus 2 curves over the field with $q$ elements can be parameterized by $3$-radicals. This results in the existence of a deterministic encoding into these curves when $q$ is congruent to $2$ modulo $3$. We extend this construction to parameterizations by $l$-radicals for small odd integers $l$, and make it explicit for $l=5$.
Citation: Jean-Marc Couveignes, Reynald Lercier. The geometry of some parameterizations and encodings. Advances in Mathematics of Communications, 2014, 8 (4) : 437-458. doi: 10.3934/amc.2014.8.437
References:
[1]

Amer. J. Math., 10 (1887), 47-70. doi: 10.2307/2369402.  Google Scholar

[2]

in Adv. Crypt. - CRYPTO' 2001 (ed. J. Kilian), Springer-Verlag, Berlin, 2001, 213-229. doi: 10.1007/3-540-44647-8_13.  Google Scholar

[3]

J. London Math. Soc., 64 (2001), 29-43. doi: 10.1017/S0024610701002113.  Google Scholar

[4]

Abh. der k. Ges. Wiss. zu Göttingen, 14 (1869), 17-75. Google Scholar

[5]

J. Symb. Comput., 47 (2012), 266-281. doi: 10.1016/j.jsc.2011.11.003.  Google Scholar

[6]

N. Elkies, The identification of three moduli spaces,, preprint, ().   Google Scholar

[7]

in Africa CRYPT, 2011, 278-289. doi: 10.1007/978-3-642-21969-6_17.  Google Scholar

[8]

in Pairing-Based Cryptography (eds. M. Joye, A. Miyaji and A. Otsuka), Springer, 2010, 265-277. doi: 10.1007/978-3-642-17455-1_17.  Google Scholar

[9]

in Graphs and Algorithms, 1989, 61-79. doi: 10.1090/conm/089/1006477.  Google Scholar

[10]

J. Symb. Comp., 51 (2013), 3-21. doi: 10.1016/j.jsc.2012.03.004.  Google Scholar

[11]

in CRYPTO, 2009, 303-316. doi: 10.1007/978-3-642-03356-8_18.  Google Scholar

[12]

Ann. Math., 72 (1960), 612-649. doi: 10.2307/1970233.  Google Scholar

[13]

in Pairing, 2010, 278-297. doi: 10.1007/978-3-642-17455-1_18.  Google Scholar

[14]

Springer, 2002. doi: 10.1007/978-1-4613-0041-0.  Google Scholar

[15]

in ANTS X - Proc. 10th Algor. Number Theory Symp. (eds. E.W. Howe and K.S. Kedlaya), Math. Sci. Publ., 2013, 463-486. Google Scholar

[16]

Springer-Verlag, Berlin, 2000.  Google Scholar

[17]

Chelsea Publishing Co., New York, 1885. Google Scholar

[18]

Bull. Pol. Acad. Sci. Math., 52 (2004), 223-226. doi: 10.4064/ba52-3-1.  Google Scholar

[19]

Acta Arith., 117 (2005), 293-301. doi: 10.4064/aa117-3-7.  Google Scholar

[20]

in Algorithmic Number Theory, Springer, Berlin, 2006, 510-524. doi: 10.1007/11792086_36.  Google Scholar

[21]

Second edition, Springer-Verlag, Berlin, 2009.  Google Scholar

[22]

Bull. Polish Acad. Sci. Math., 55 (2007), 97-104. doi: 10.4064/ba55-2-1.  Google Scholar

show all references

References:
[1]

Amer. J. Math., 10 (1887), 47-70. doi: 10.2307/2369402.  Google Scholar

[2]

in Adv. Crypt. - CRYPTO' 2001 (ed. J. Kilian), Springer-Verlag, Berlin, 2001, 213-229. doi: 10.1007/3-540-44647-8_13.  Google Scholar

[3]

J. London Math. Soc., 64 (2001), 29-43. doi: 10.1017/S0024610701002113.  Google Scholar

[4]

Abh. der k. Ges. Wiss. zu Göttingen, 14 (1869), 17-75. Google Scholar

[5]

J. Symb. Comput., 47 (2012), 266-281. doi: 10.1016/j.jsc.2011.11.003.  Google Scholar

[6]

N. Elkies, The identification of three moduli spaces,, preprint, ().   Google Scholar

[7]

in Africa CRYPT, 2011, 278-289. doi: 10.1007/978-3-642-21969-6_17.  Google Scholar

[8]

in Pairing-Based Cryptography (eds. M. Joye, A. Miyaji and A. Otsuka), Springer, 2010, 265-277. doi: 10.1007/978-3-642-17455-1_17.  Google Scholar

[9]

in Graphs and Algorithms, 1989, 61-79. doi: 10.1090/conm/089/1006477.  Google Scholar

[10]

J. Symb. Comp., 51 (2013), 3-21. doi: 10.1016/j.jsc.2012.03.004.  Google Scholar

[11]

in CRYPTO, 2009, 303-316. doi: 10.1007/978-3-642-03356-8_18.  Google Scholar

[12]

Ann. Math., 72 (1960), 612-649. doi: 10.2307/1970233.  Google Scholar

[13]

in Pairing, 2010, 278-297. doi: 10.1007/978-3-642-17455-1_18.  Google Scholar

[14]

Springer, 2002. doi: 10.1007/978-1-4613-0041-0.  Google Scholar

[15]

in ANTS X - Proc. 10th Algor. Number Theory Symp. (eds. E.W. Howe and K.S. Kedlaya), Math. Sci. Publ., 2013, 463-486. Google Scholar

[16]

Springer-Verlag, Berlin, 2000.  Google Scholar

[17]

Chelsea Publishing Co., New York, 1885. Google Scholar

[18]

Bull. Pol. Acad. Sci. Math., 52 (2004), 223-226. doi: 10.4064/ba52-3-1.  Google Scholar

[19]

Acta Arith., 117 (2005), 293-301. doi: 10.4064/aa117-3-7.  Google Scholar

[20]

in Algorithmic Number Theory, Springer, Berlin, 2006, 510-524. doi: 10.1007/11792086_36.  Google Scholar

[21]

Second edition, Springer-Verlag, Berlin, 2009.  Google Scholar

[22]

Bull. Polish Acad. Sci. Math., 55 (2007), 97-104. doi: 10.4064/ba55-2-1.  Google Scholar

[1]

Lei Lei, Wenli Ren, Cuiling Fan. The differential spectrum of a class of power functions over finite fields. Advances in Mathematics of Communications, 2021, 15 (3) : 525-537. doi: 10.3934/amc.2020080

[2]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021063

[3]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[4]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031

[5]

Habib Ammari, Josselin Garnier, Vincent Jugnon. Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 389-417. doi: 10.3934/dcdss.2015.8.389

[6]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[7]

Zheng Liu, Tianxiao Wang. A class of stochastic Fredholm-algebraic equations and applications in finance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3879-3903. doi: 10.3934/dcdsb.2020267

[8]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141

[9]

Thomas Kappeler, Yannick Widmer. On nomalized differentials on spectral curves associated with the sinh-Gordon equation. Journal of Geometric Mechanics, 2021, 13 (1) : 73-143. doi: 10.3934/jgm.2020023

[10]

Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021047

[11]

Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021037

[12]

Azmeer Nordin, Mohd Salmi Md Noorani. Counting finite orbits for the flip systems of shifts of finite type. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021046

[13]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

[14]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

[15]

Andrés Contreras, Juan Peypouquet. Forward-backward approximation of nonlinear semigroups in finite and infinite horizon. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021051

[16]

Tianhu Yu, Jinde Cao, Chuangxia Huang. Finite-time cluster synchronization of coupled dynamical systems with impulsive effects. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3595-3620. doi: 10.3934/dcdsb.2020248

[17]

Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem. Electronic Research Archive, , () : -. doi: 10.3934/era.2021031

[18]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265

[19]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[20]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

2019 Impact Factor: 0.734

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]