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from sage.all import *
def torsion_to_theta_null_point(P):
r=P[0]
s=P[2]
return (r+s,r-s)
def montgomery_point_to_theta_point(O,P):
if P[0]==P[2]==0:
return O
else:
a,b=O
return (a*(P[0]-P[2]),b*(P[0]+P[2]))
def theta_point_to_montgomery_point(E,O,P,twist=False):
a,b=O
x=a*P[1]+b*P[0]
z=a*P[1]-b*P[0]
if twist:
x=-x
if z==0:
return E(0)
else:
x=x/z
a_inv=E.a_invariants()
A =a_inv[1]
if a_inv != (0,A,0,1,0):
raise ValueError("The elliptic curve E is not in the Montgomery model.")
y2=x*(x**2+A*x+1)
if not is_square(y2):
raise ValueError("The Montgomery point is not on the base field.")
else:
y=y2.sqrt()
return E([x,y,1])
def lift_kummer_montgomery_point(E,x,z=1):
if z==0:
return E(0)
elif z!=1:
x=x/z
a_inv=E.a_invariants()
A =a_inv[1]
if a_inv != (0,A,0,1,0):
raise ValueError("The elliptic curve E is not in the Montgomery model.")
y2=x*(x**2+A*x+1)
if not is_square(y2):
raise ValueError("The Montgomery point is not on the base field.")
else:
y=y2.sqrt()
return E([x,y,1])
def null_point_to_montgomery_coeff(a,b):
return -2*(a**4+b**4)/(a**4-b**4)
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