Sato Tate and notions of generality in cryptography

Description

Sato-Tate and notions of generality in cryptography David R. Kohel Institut de Mathematiques de Luminy Geocrypt 2011, Corsica, 20 June 2011

• point-counting algorithm


• family over


• dimensional cm


• dimensional family


• frobenius angles


• galois distributions


• arithmetic speciality

Sujets

• multiplication
• maths

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 17
• Langue: English

Extrait

Sato-Tate and notions of generality
in cryptography
David R. Kohel
Institut de Mathematiques de Luminy
Geocrypt 2011, Corsica, 20 June 2011Families of curves in cryptography
We considerC!S a family of curves, such that each ber over a
closed point x of S is a curve C=k =F . In cryptographicq
applications we are interested in the properties of J = Jac(C) as
we vary x in S.
Examples. The rst examples are elliptic curves.
2 31.E :y =x +ax +b over S, where
1 12S = Spec(Z[a;b; ])A =Z[ ];
6ab 6
a family of dimension 3.
2 3 22.E :y +xy =x +ax +b=S where
1 2S = Spec(F [a;b; ])A =F ;2 2
b
a family of dimension 2.Examples of cryptographic curve families
2 3 23.E :y =x +x 3x + 1=S, where
1
S = Spec(Z[ ]);
2
p
a CM family with endomorphism ringZ[ 2], of dimension 1.
Next we consider families of genus 2 curves.
2 5 34.C :y =x + 5x + 5x +t over S, where
1 11S = Spec(Z[t; ])A =Z[ ];
230(t + 4) 30
p
a 2-dimensional family with real multiplication by Z[(1 + 5)=2]
for which we will present an e cient point-counting algorithm.
2 55.C :y =x + 1, a one-dimensional CM family over
1
S = Spec(Z[ ]):
10Notions of generality in cryptography
We address the question: "What is special about special curves?"
The notion of speciality can be separated into the geometric and
arithmetic properties.
Geometric speciality. IfC!S is a family (of genus g curves),
what is the induced image S!X in the moduli space (inM ).g
Arithmetic speciality. Here we distiguish the (local) level
structure and the (global or geometric) Galois distributions.
a. What level structure is xed by the family? | Is there an
exceptional N such that the Galois representation
: Gal(Q=Q)! GL (Z=NZ)N 2g
is smaller than expected?
b. What is the image of the Galois action on the Tate module?
: Gal(Q=Q)! Aut(T (J)) = GL (Z ):‘ ‘ 2g ‘Frobenius angles and normalized traces
Let E=Q be an elliptic curve, with discriminant , viewed as a
1scheme over S = Spec(Z[ ]). The Sato{Tate conjecture concerns

the distribution of the Frobenius angles at primes p.
For each p, let = be the Frobenius endomorphism on E=Fp p
and
2(T ) =T a T +pp
its characteristic polynomial of Frobenius. Set t equal to thep
normalized Frobenius trace
p
t =a = p;p p
and denote by in [0;] the Frobenius angle, de ned byp
i pt = 2 cos( ). We set =e (the unit Frobenius), andp p p
2e(T ) =T t T + 1 = (T )(T ):p p pSato{Tate Conjecture
Sato{Tate Conjecture. Suppose that E=Q is a non-CM elliptic
curve. For [;] [0;],
Z 2jfpNj gj 2 sin ()plim = d ;
N!1 jfpNgj
or equivalently for [a;b] [ 2; 2],
pZ b 2jfpNjat bgj 4 tp
lim = dt:
N!1 jfpNgj 2a
The analogous distributions for CM elliptic curves is classical:
Z jfpNj