I’ve been reading about Weil pairing from Pairings for Beginners and in section 5.1 an example is given. I took a look on the magma code of that example (see here) and it works.
All I did was to change the point $ Q(21,12i)$ which is in the trace zero subgroup in $ 3$ -torsion to the other point $ Q'(21,11i)$ which also is in the same subgroup. Hence the pairing is still non-degenerate.
Question. It seems that the pairing is not computed as when we call the function
WeilPairing(P,Q,r) from magma. Why is this happening ?
You could run magma scripts here. I attached the modified code (which again is just the point $ Q$ which switched to $ Q’$ ).
clear; q:=23; Fq:=GF(q); a:=-1; b:=0; E:=EllipticCurve([Fq|a,b]); P:=E![2,11]; Fq2<i>:=ExtensionField<Fq,x|x^2+1>; r:=3; E2:=BaseChange(E,Fq2); O:=PointsAtInfinity(E2); pi:=FrobeniusMap(E2); //Q:=(#E2 div r^2)*Random(E2); //(pi(Q)-q*Q) eq PointsAtInfinity(E2) and Order(Q) eq 3; P:=E2!P; Q:=E2![21,11*i]; R:=E2![17*i,2*i + 21]; S:=E2![10*i + 18,13*i + 13]; F<x,y>:=FunctionField(E2); fDBL:=function(P); lambda:=(3*P^2+a)/(2*P); c:=P-lambda*P; l:=F!(y-(lambda*x+c)); v:=F!(x-(lambda^2-2*P)); return F!(l/v); end function; fADD:=function(P,Q); lambda:=(Q-P)/(Q-P); c:=P-lambda*P; l:=F!(y-(lambda*x+c)); v:=F!(x-(lambda^2-P-P)); return F!(l/v); end function; f3P:=fDBL(P)*(x-P); lPR:=fADD(P,R); f3PR:=f3P/lPR^3; f3Q:=fDBL(Q)*(x-Q); lQS:=fADD(Q,S); f3QS:=f3Q/lQS^3; f3PR(Q+S)*f3QS(R)/(f3PR(S)*f3QS(P+R)); WeilPairing(P,Q,r); WeilPairing(2*P,Q,r); WeilPairing(P,2*Q,r);
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