Elliptic Curves Number Theory And Cryptography——pairing learning

《Elliptic Curves Number Theory And Cryptography 2n》中Example 11.5, magma脚本：

clear;
q:=7;
Fq:=GF(q);
Fq2:=ExtensionField;
a:=0;
b:=2;
E:=EllipticCurve([Fq|a,b]);
fac:=Factorization(#E);
fac2:=Factorization(#E(Fq2));
points:=Points(E(Fq2));

TorsPts:={};

for i:=1 to #points do
	if 3*points[i] eq Identity(E) then
		Include(~TorsPts,points[i]);
	end if;
end for;

function FlowerGenerator(TorsPts);
	V:={};
	S:=TorsPts;
	petals:=Integers()!Sqrt(#S)+1;
	ptsInPet:=petals-1;
	T:={};

	for i:=1 to petals do
		S:=S diff V;
  		P:=Random(S);
	        while P[3] eq 0 do
			P:=Random(S);
  		end while;
	V:={j*P:j in [1..ptsInPet]};
 	Include(~T,V);
	end for;
return T;
end function;

FlowerGenerator(TorsPts);

P:=E(Fq)![5,1];
Q:=E(Fq)![6,1];
R:=E(Fq)![0,3];
P+Q;
identitiy:=E(Fq)![0,1,0];
P+Q-P-Q;
identitiy;

对应执行结果为：

{
    { (0 : 1 : 0), (0 : 4 : 1), (0 : 3 : 1) },
    { (0 : 1 : 0), (5 : 6 : 1), (5 : 1 : 1) },
    { (6 : 1 : 1), (0 : 1 : 0), (6 : 6 : 1) },
    { (3 : 6 : 1), (0 : 1 : 0), (3 : 1 : 1) }
}
(3 : 6 : 1)
(0 : 1 : 0)
(0 : 1 : 0)

Example 11.7对应magma脚本为：

clear;
q:=11;
Fq:=GF(q);
Fq2:=ExtensionField;
a:=-1;
b:=1;
E:=EllipticCurve([Fq|a,b]);
fac:=Factorization(#E);
fac2:=Factorization(#E(Fq2));
points:=Points(E(Fq2));

#E;
Points(E(Fq));

TorsPts:={};

for i:=1 to #points do
	if 5*points[i] eq Identity(E) then
		Include(~TorsPts,points[i]);
	end if;
end for;

TorsPts;

P:=E(Fq)![3,6];
Q:=E(Fq)![3,6];
R:=E(Fq)![0,1];
P+Q;
4*P;
5*P;
identitiy:=E(Fq)![0,1,0];
P+Q-P-Q;
identitiy;
3^5 mod 11;

对应执行结果为：

10
{@ (0 : 1 : 0), (0 : 1 : 1), (0 : 10 : 1), (1 : 1 : 1), (1 : 10 : 1), (3 : 5 :
1), (3 : 6 : 1), (5 : 0 : 1), (10 : 1 : 1), (10 : 10 : 1) @}
{ (10 : 10 : 1), (3 : 5 : 1), (0 : 1 : 0), (10 : 1 : 1), (3 : 6 : 1) }
(10 : 10 : 1)
(3 : 5 : 1)
(0 : 1 : 0)
(0 : 1 : 0)
(0 : 1 : 0)
1