1 files changed, 76 insertions, 0 deletions
diff --git a/theta_lib/basis_change/canonical_basis_dim1.py b/theta_lib/basis_change/canonical_basis_dim1.py
new file mode 100644
index 0000000..e1c3d1f
--- /dev/null
+++ b/theta_lib/basis_change/canonical_basis_dim1.py
@@ -0,0 +1,76 @@
+from sage.all import *
+from ..utilities.discrete_log import weil_pairing_pari, discrete_log_pari
+
+def last_four_torsion(E):
+	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=A-2
+	y=y2.sqrt()
+	return E([-1,y,1])
+
+
+def make_canonical(P,Q,A,preserve_pairing=False):
+	r"""
+	Input:
+	- P,Q: a basis of E[A].
+	- A: an integer divisible by 4.
+	- preserve_pairing: boolean indicating if we want to preserve pairing at level 4.
+	
+	Output:
+	- P1,Q1: basis of E[A].
+	- U1,U2: basis of E[4] induced by (P1,Q1) ((A//4)*P1=U1, (A//4)*Q1=U2) such that U2[0]=-1 
+	and e_4(U1,U2)=i if not preserve_pairing and e_4(U1,U2)=e_4((A//4)*P,(A//4)*Q) if preserve_pairing.
+	- M: base change matrix (in row convention) from (P1,Q1) to (P,Q).
+	
+	We say that (U1,U2) is canonical and that (P1,Q1) induces or lies above a canonical basis. 
+	"""
+	E=P.curve()
+	Fp2=E.base_ring()
+	i=Fp2.gen()
+
+	assert i**2==-1
+
+	T2=last_four_torsion(E)
+	V1=(A//4)*P
+	V2=(A//4)*Q
+	U1=V1
+	U2=V2
+	
+	a1=discrete_log_pari(weil_pairing_pari(U1,T2,4),i,4)
+	b1=discrete_log_pari(weil_pairing_pari(U2,T2,4),i,4)
+
+	if a1%2!=0:
+		c1=inverse_mod(a1,4)
+		d1=c1*b1
+		P1=P
+		Q1=Q-d1*P
+		U1,U2=U1,U2-d1*U1
+		M=matrix(ZZ,[[1,0],[d1,1]])
+	else:
+		c1=inverse_mod(b1,4)
+		d1=c1*a1
+		P1=Q
+		Q1=P-d1*Q
+		U1,U2=U2,U1-d1*U2
+		M=matrix(ZZ,[[d1,1],[1,0]])
+
+	if preserve_pairing:
+		e4=weil_pairing_pari(V1,V2,4)
+	else:
+		e4=i
+	
+	if weil_pairing_pari(U1,U2,4)!=e4:
+		U2=-U2
+		Q1=-Q1
+		M[0,1]=-M[0,1]
+		M[1,1]=-M[1,1]
+	
+	assert (A//4)*P1==U1
+	assert (A//4)*Q1==U2
+	assert weil_pairing_pari(U1,U2,4)==e4
+	assert M[0,0]*P1+M[0,1]*Q1==P
+	assert M[1,0]*P1+M[1,1]*Q1==Q
+	
+	return P1,Q1,U1,U2,M