halving points on elliptic curve

[somayeh didari <somayeh_didari@yahoo.com>]

thanks for your help, I wrote this program, which take elliptic curve e over the rational points and a point Q in e and returns (1/2)Q:

halve(e,Q)={

x_0=Q[1];

Half=[];

f=x^4-e.b4*x^2-2*e.b6*x-e.b8-x_0*(4*x^3+e.b2*x^2+2*e.b4*x+e.b6);

g=factor(f);

v=[];

  for(i=1,#g~,

if(poldegree(g[i,1])==1,

v=concat(v,-polcoeff(g[i,1],0)/polcoeff(g[i,1],1));

    );

);

for(i=1,#v~,

x=v[i];

y=ellordinate(e,x)[1];

if(ellpow(e,[x,y],2)==Q,

Half=concat(Half,[[x,y]]);

);

if(ellpow(e,[x,-1*y],2)==Q,

Half=concat(Half,[[x,-1*y]]);

);

);

}

It works! But there is a problem, when I use

e=ellinit([0,0,0,-1563056672958141,0]);

P_1=[48408867,194361588954];

P_2=[48432972,194700535386];

Q=elladd(e,ellpow(e,P_1,2),ellpow(e,P_2,3));

halve(e,Q);

Then Half will be [], as I want. but when I use

Q1=ellpow(e,Q,2)

halve(e,Q1)

Q1=Half[1]

halve(e,Q1);

Pari doesn't work. But we now Q1=Q! Why???????Is there an algorithm which computes (1/2)Q?

 

somayeh didari