Smaller class invariants for constructing curves of genus

Marco Streng

Description

Smaller class invariants for constructing curves of genus 2 Marco Streng Geocrypt Corsica June 2011

• let j0 ?


• class polynomial


• smaller class


• invariants part


• hilbert class polynomial

Sujets

• multiplication

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 14
• Langue: English

Extrait

for

Smaller class invariants constructing curves of genus

Marco

Streng

Geocrypt Corsica June 2011

2

Overview

constructing curves

smaller class invariants

genus 1

part

part

1

3

genus 2

part

part

2

4

Part 1: The Hilbert class polynomial

Definition:Thej -invariantis

j(E) 2 2 b 2 8 3 3 + 3 b 3 3 3 c 2 forE:y 2 =x 3 +bx+c. = Fact:j(E) =j(F)⇐⇒E ∼ = k F

Definition:LetKbe an imaginary quadratic number field. ItsHilbert class polynomialis H K = Y  X−j(E)  ∈Z[X]. E/C End(E) ∼ O K = Application 1:roots generate Hilbert class field ofK Application 2:elliptic curves with prescribed Frobenius

Elliptic curves with prescribed Frobenius

Algorithm:(givenπ∈ O K imag. quadr. withp=ππprime) 1.ComputeH K modp, it splits into linear factors. 2.Letj 0 ∈F p be a root and letE 0 /F p havej(E 0 ) =j 0 .

3.

Select

the

twistEofE 0 with “Frob =π”. It satisfies

#E(F p ) =N(π−1) =p+ 1−tr(π).