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November  2014, 8(4): 375-387. doi: 10.3934/amc.2014.8.375

Trisection for supersingular genus $2$ curves in characteristic $2$

Josep M. Miret 1, , Jordi Pujolàs 1, and  Nicolas Thériault 2,

1. 

Departament de Matemàtica, Universitat de Lleida, Jaume II 69, Lleida 25001, Spain
2. 

Departamento de Matemática, Universidad del Bío-Bío, Avenida Collao 1202, Concepción, Chile

Received  January 2014 Revised  June 2014 Published  November 2014

By reversing reduction in divisor class arithmetic we provide efficient trisection algorithms for supersingular Jacobians of genus $2$ curves over finite fields of characteristic $2$. With our technique we obtain new results for these Jacobians: we show how to find their $3$-torsion subgroup, we prove there is none with $3$-torsion subgroup of rank $3$ and we prove that the maximal $3$-power order subgroup is isomorphic to either $\mathbb{Z}/3^{v}\mathbb{Z}$ or $(\mathbb{Z}/3^{\frac v2}\mathbb{Z})^2$ or $(\mathbb{Z}/3^{\frac v4}\mathbb{Z})^4$, where $v$ is the $3$-adic valuation $v_{3}$(#Jac(C)$(\mathbb{F}_{2^m})$). Ours are the first trisection formulae available in literature.
Keywords: Hyperelliptic curve, explicit formulas., binary field, trisection, supersingular, divisor class, genus 2.
Mathematics Subject Classification: Primary: 11G20, 14H40, 14H4.
Citation: Josep M. Miret, Jordi Pujolàs, Nicolas Thériault. Trisection for supersingular genus $2$ curves in characteristic $2$. Advances in Mathematics of Communications, 2014, 8 (4) : 375-387. doi: 10.3934/amc.2014.8.375
References:
[1]

D. Cantor, Computing in the Jacobian of a Hyperelliptic curve, Math. Comp., 48 (1987), 95-101. doi: 10.1090/S0025-5718-1987-0866101-0.  Google Scholar

[2]

I. Kitamura, M. Katagi and T. Takagi, A complete divisor class halving algorithm for hyperelliptic curve cryptosystems of genus two, in Information Security and Privacy, Springer, 2005, 146-157. doi: 10.1007/11506157_13.  Google Scholar

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J. Miret, J. Pujolàs and A. Rio, Explicit 2-power torsion of genus $2$ curves over finite fields, Adv. Math. Commun., 4 (2010), 155-165. doi: 10.3934/amc.2010.4.155.  Google Scholar

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F. Oort, Subvarieties of moduli spaces, Invent. Math., 24 (1974), 95-119. doi: 10.1007/BF01404301.  Google Scholar

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R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, 46 (1987), 183-211. doi: 10.1016/0097-3165(87)90003-3.  Google Scholar

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C. Xing, On supersingular abelian varieties of dimension two over finite fields, Finite Fields Appl., 2 (1996), 407-421. doi: 10.1006/ffta.1996.0024.  Google Scholar

show all references

References:
[1]

D. Cantor, Computing in the Jacobian of a Hyperelliptic curve, Math. Comp., 48 (1987), 95-101. doi: 10.1090/S0025-5718-1987-0866101-0.  Google Scholar

[2]

I. Kitamura, M. Katagi and T. Takagi, A complete divisor class halving algorithm for hyperelliptic curve cryptosystems of genus two, in Information Security and Privacy, Springer, 2005, 146-157. doi: 10.1007/11506157_13.  Google Scholar

[3]

J. Miret, J. Pujolàs and A. Rio, Explicit 2-power torsion of genus $2$ curves over finite fields, Adv. Math. Commun., 4 (2010), 155-165. doi: 10.3934/amc.2010.4.155.  Google Scholar

[4]

F. Oort, Subvarieties of moduli spaces, Invent. Math., 24 (1974), 95-119. doi: 10.1007/BF01404301.  Google Scholar

[5]

R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A, 46 (1987), 183-211. doi: 10.1016/0097-3165(87)90003-3.  Google Scholar

[6]

C. Xing, On supersingular abelian varieties of dimension two over finite fields, Finite Fields Appl., 2 (1996), 407-421. doi: 10.1006/ffta.1996.0024.  Google Scholar

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