1 Etymology
2 Case of elliptic curves
3 See also
4 References

[edit] Etymology
From the Greek (iso-) and Latin (genus), the term isogeny means "equal origins", a reference to the geometrical fact that an isogeny sends the point at infinity (the origin) of the source elliptic curve to the point at infinity of the target elliptic curve.

[edit] Case of elliptic curves
For elliptic curves, this notion can also be formulated as follows:

Let E1 and E2 be elliptic curves over a field k. An isogeny between E1 and E2 is a surjective morphism  of varieties that preserves basepoints (i.e. f maps the infinite point on E1 to that on E2).

Two elliptic curves E1 and E2 are called isogenous if there is an isogeny . This is an equivalence relation, symmetry being due to the existence of the dual isogeny. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the elliptic curves.

椭圆曲线自同态《信息安全中的数学方法与技术》冯登国书里举了4个例子

1:
p mod4==1，iy的i要求是4阶点，随机验了5个点都没问题：
y^2=x^3+21x

f: (x,y)------>(-x,iy)

-x^3-21x      =     i^2*y^2=-y^2

p:=NextPrime (13);
p;
pp:=p mod 4;
pp;
K := FiniteField(p);
K;
E := EllipticCurve([K|0,0,0,21,0]);
E;
v:=# E;
v;
FactoredOrder(E) ;
Trace(E);

TraceOfFrobenius(E);

r1:=Random(E);

r1;
r2:=Random(E);
r2;

Points(E);
r1+r2;
m1:=Points(E, 451);
m1;
m2:=Points(E, -451);
m2;
IsPoint(E, 451);
h:=12^3+4*12 ;
h;
hh:= 5^2 ;
hh;
(h mod p) eq (hh mod p);
g:=(-12^3+4*(-12) )mod p;
g;
z1:=E![5,3];
z2:=E![2,13];
z3:=E![6,11];
z4:=E![11,7];
z5:=E![15,16];
gg:=5 * z1 ;
gg;

gg eq z1;
ggg:=5 * z2 ;
ggg;
ggg eq z2;
gggg:=5 * z3;
gggg;
gggg eq z3;

ggggg:= 5 *z4;
ggggg;
gggggg:=5 *z5;
gggggg eq z5;
Order(r1);
Order(r2);

17
1
Finite field of size 17
Elliptic Curve defined by y^2 = x^3 + 4*x over GF(17)

16
[ <2, 4> ]
2
2
(12 : 5 : 1)
(2 : 13 : 1)
{@ (0 : 1 : 0), (0 : 0 : 1), (2 : 4 : 1), (2 : 13 : 1), (5 : 3 : 1), (5 : 14 :
1), (6 : 6 : 1), (6 : 11 : 1), (8 : 0 : 1), (9 : 0 : 1), (11 : 7 : 1), (11 : 10
: 1), (12 : 5 : 1), (12 : 12 : 1), (15 : 1 : 1), (15 : 16 : 1) @}
(5 : 3 : 1)
[ (9 : 0 : 1) ]
[ (8 : 0 : 1) ]
true (9 : 0 : 1)
1776
25
true
9
(5 : 3 : 1)
true
(2 : 13 : 1)
true
(6 : 11 : 1)
true
(11 : 7 : 1)
true
4
4

2:

p mod 3  要找3阶元乘横座标

f: (x,y)------>(kx,y)

特征：f^2+f+1=0的根为ω，ω^3=1,ω^2=1-ω

ω^3*x^3+5=x^3+5=y^2

p:=NextPrime (17);
p;
pp:=p mod 3;
pp;
K := FiniteField(p);
K;
E := EllipticCurve([K|0,0,0,0,5]);
E;
v:=# E;
v;
FactoredOrder(E) ;
Trace(E);

TraceOfFrobenius(E);

r1:=Random(E);

r1;
r2:=Random(E);
r2;

Points(E);

z1:=E![8,17];
z1;
gg:=18*z1;
gg ;
ggg:=16*z1;
ggg;
gggg:=7*z1;
gggg;
ggggg:=6*z1;
ggggg;
gggggg:=10*z1;
gggggg;
ggggggg:=12*z1;
ggggggg;
gggggggg:=1*z1;
gggggggg;
ggggggggg:=0*z1;
ggggggggg;

Order(r1);
Order(r2);

19
1
Finite field of size 19
Elliptic Curve defined by y^2 = x^3 + 5 over GF(19)
27
[ <3, 3> ]
-7
-7
(1 : 14 : 1)
(12 : 17 : 1)
{@ (0 : 1 : 0), (0 : 9 : 1), (0 : 10 : 1), (1 : 5 : 1), (1 : 14 : 1), (5 : 4 :
1), (5 : 15 : 1), (7 : 5 : 1), (7 : 14 : 1), (8 : 2 : 1), (8 : 17 : 1), (10 : 6
: 1), (10 : 13 : 1), (11 : 5 : 1), (11 : 14 : 1), (12 : 2 : 1), (12 : 17 : 1),
(13 : 6 : 1), (13 : 13 : 1), (15 : 6 : 1), (15 : 13 : 1), (16 : 4 : 1), (16 : 15
: 1), (17 : 4 : 1), (17 : 15 : 1), (18 : 2 : 1), (18 : 17 : 1) @}
(8 : 17 : 1)
(0 : 1 : 0)
(8 : 17 : 1)
(8 : 17 : 1)
(0 : 1 : 0)
(8 : 17 : 1)
(0 : 1 : 0)
(8 : 17 : 1)
(0 : 1 : 0)9
3

3

f:(x,y)---->f(x^q,y^q)

特征:f^2-t*x+q=0,t为FrobeniusMap迹，f^2-t*x+q=0的根为[-t+(t^2-4q)^(1/2)]/2和[-t-(t^2-4q)^(1/2)]/2

一个很难的定理：Frobenius映射叠代定理得出：
x^(3q)+a*x^q+b=y^3

K := FiniteField(2,101); // finite field of size 2^101
> E := EllipticCurve([K!1,0,0,0,1]);
E;
# E;
NumberOfGenerators(E);
AutomorphismGroup(E);
FactoredOrder(E) ;
FrobeniusMap(E);
FrobeniusMap(E, 3);
Trace(E);

E1 := EllipticCurve([ GF(23) | 1, 1 ]);
> E2 := EllipticCurve([ GF(29, 2) | 1, 1 ]);
f := map<E1 -> E2 | P :-> 2*P>;f;
不同域之间，只要a，b同就可能找出同态

Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^101)
2535301200456455833701195805484
1
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Power Structure of MapSch given by a rule [no inverse]
[ <2, 2>, <1211597, 1>, <523132114155213291569143, 1> ]
Elliptic curve isogeny from: CrvEll: E to Elliptic Curve defined by y^2 + x*y =
x^3 + 1 over GF(2^101)
taking (x : y : 1) to (x^2 : y^2 : 1)
Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^101)
Elliptic curve isogeny from: CrvEll: E to Elliptic Curve defined by y^2 + x*y =
x^3 + 1 over GF(2^101)
taking (x : y : 1) to (x^8 : y^8 : 1)
Elliptic Curve defined by y^2 + x*y = x^3 + 1 over GF(2^101)
2969292210605269
Mapping from: CrvEll: E1 to CrvEll: E2 given by a rule [no inverse]

http://jpkc.szpt.edu.cn/2007/sznk/UploadFile/ziliao/wenzhang/smartcard/ECC%D4%DA%D6%C7%C4%DC%BF%A8%C9%CF%B5%C4%CA%B5%CF%D6.PDF

http://www.math.uga.edu/~pete/SC1-endomorphisms.pdf

Computing the endomorphism ring of an
ordinary elliptic curve

上传的附件：

• 自同态的次数为3的椭圆曲线_英文_.pdf （451.06kb，18次下载）
• CCRTalk.pdf （131.51kb，2次下载）