Initial Commit

Co-Authored-By: Damien Robert <Damien.Olivier.Robert+git@gmail.com>
Co-Authored-By: Frederik Vercauteren <frederik.vercauteren@gmail.com>
Co-Authored-By: Jonathan Komada Eriksen <jonathan.eriksen97@gmail.com>
Co-Authored-By: Pierrick Dartois <pierrickdartois@icloud.com>
Co-Authored-By: Riccardo Invernizzi <nidadoni@gmail.com>
Co-Authored-By: Ryan Rueger <git@rueg.re>                                                                                                                                           [0.01s]
Co-Authored-By: Benjamin Wesolowski <benjamin@pasch.umpa.ens-lyon.fr>
Co-Authored-By: Arthur Herlédan Le Merdy <ahlm@riseup.net>
Co-Authored-By: Boris Fouotsa <tako.fouotsa@epfl.ch>

1 files changed, 567 insertions, 0 deletions

diff --git a/theta_lib/isogenies/Kani_gluing_isogeny_chain_dim4.py b/theta_lib/isogenies/Kani_gluing_isogeny_chain_dim4.py
new file mode 100644
index 0000000..282219c
--- /dev/null
+++ b/theta_lib/isogenies/Kani_gluing_isogeny_chain_dim4.py
@@ -0,0 +1,567 @@
+from sage.all import *
+from ..utilities.discrete_log import weil_pairing_pari
+from ..basis_change.canonical_basis_dim1 import make_canonical
+from ..basis_change.kani_base_change import (
+    fixed_deg_gluing_matrix_Phi1,
+    fixed_deg_gluing_matrix_Phi2,
+    fixed_deg_gluing_matrix_dim4,
+    clapoti_cob_matrix_dim2,
+    clapoti_cob_matrix_dim2_dim4,
+    gluing_base_change_matrix_dim2,
+    gluing_base_change_matrix_dim2_dim4,
+    gluing_base_change_matrix_dim2_F1,
+    gluing_base_change_matrix_dim2_F2,
+    kernel_basis,
+)
+from ..basis_change.base_change_dim2 import base_change_theta_dim2
+from ..basis_change.base_change_dim4 import base_change_theta_dim4
+from ..theta_structures.Theta_dim1 import ThetaStructureDim1
+from ..theta_structures.Theta_dim2 import ProductThetaStructureDim2
+from ..theta_structures.Tuple_point import TuplePoint
+from ..theta_structures.Theta_dim4 import ProductThetaStructureDim2To4, ThetaPointDim4
+from ..isogenies_dim2.isogeny_chain_dim2 import IsogenyChainDim2
+from .gluing_isogeny_dim4 import GluingIsogenyDim4
+
+class KaniFixedDegDim2Gluing:
+	def __init__(self,P_mp3,Q_mp3,a,b,c,d,u,f,m,strategy_dim2=None):
+		r"""
+		INPUT:
+		- P_mp3, Q_mp3: basis of E[2^(m+3)] such that pi(P_mp3)=P_mp3 and pi(Q_mp3)=-Q_mp3.
+		- a,b,c,d,u,f: integers such that a**2+c**2+p*(b**2+d**2)=u*(2**f-u), where p is
+		ths characteristic of the base field.
+		- m: integer such that m=min(v_2(a-b),v_2(a+b)).
+
+		OUTPUT: Gluing isogeny chain F_{m+1}\circ...\circ F_1 containing the first m+1 steps of
+		the isogeny F: E^4 --> A*A' representing a u-isogeny in dimension 2.
+		"""
+
+		P_mp2 = 2*P_mp3
+		Q_mp2 = 2*Q_mp3
+		P_4 = 2**m*P_mp2
+		Q_4 = 2**m*Q_mp2
+
+		E = P_mp3.curve()
+
+		# Canonical basis with S_4=(1,0)
+		_, _, R_4, S_4, M_dim1 = make_canonical(P_4,Q_4,4,preserve_pairing=True)
+
+		Z4 = Integers(4)
+		M0 = matrix(Z4,[[M_dim1[0,0],0,M_dim1[0,1],0],
+			[0,M_dim1[0,0],0,M_dim1[0,1]],
+			[M_dim1[1,0],0,M_dim1[1,1],0],
+			[0,M_dim1[1,0],0,M_dim1[1,1]]])
+
+		# Theta structures
+		Theta_E = ThetaStructureDim1(E,R_4,S_4)
+		Theta_EE = ProductThetaStructureDim2(Theta_E,Theta_E)
+
+		# Gluing change of basis in dimension 2
+		M1 = fixed_deg_gluing_matrix_Phi1(u,a,b,c,d)
+		M2 = fixed_deg_gluing_matrix_Phi2(u,a,b,c,d)
+
+		M10 = M0*M1
+		M20 = M0*M2
+
+		Fp2 = E.base_field()
+		e4 = Fp2(weil_pairing_pari(R_4,S_4,4))
+
+		N_Phi1 = base_change_theta_dim2(M10,e4)
+		N_Phi2 = base_change_theta_dim2(M20,e4)
+
+		# Gluing change of basis dimension 2 * dimension 2 --> dimension 4
+		M_dim4 = fixed_deg_gluing_matrix_dim4(u,a,b,c,d,m)
+
+		self.N_dim4 = base_change_theta_dim4(M_dim4,e4)
+
+		# Kernel of Phi1 : E^2 --> A_m1 and Phi2 : E^2 --> A_m2
+		two_mp2 = 2**(m+2)
+		two_mp3 = 2*two_mp2
+		mu = inverse_mod(u,two_mp3)
+		
+		B_K_Phi1 = [TuplePoint((u%two_mp2)*P_mp2,((c+d)%two_mp2)*P_mp2),
+		TuplePoint((((d**2-c**2)*mu)%two_mp2)*Q_mp2,((c-d)%two_mp2)*Q_mp2)]
+
+		B_K_Phi2 = [TuplePoint((u%two_mp2)*P_mp2,((d-c)%two_mp2)*P_mp2),
+		TuplePoint((((d**2-c**2)*mu)%two_mp2)*Q_mp2,(-(c+d)%two_mp2)*Q_mp2)]
+
+		# Computation of the 2**m-isogenies Phi1 and Phi2
+		self._Phi1=IsogenyChainDim2(B_K_Phi1,Theta_EE,N_Phi1,m,strategy_dim2)
+		self._Phi2=IsogenyChainDim2(B_K_Phi2,Theta_EE,N_Phi2,m,strategy_dim2)
+
+		# Kernel of the (m+1)-th isogeny in dimension 4 F_{m+1}: A_m1*A_m2 --> B (gluing isogeny)
+
+		B_K_dim4 =[TuplePoint((u%two_mp3)*P_mp3,E(0),((a+b)%two_mp3)*P_mp3,((c+d)%two_mp3)*P_mp3),
+		TuplePoint(E(0),(u%two_mp3)*P_mp3,((d-c)%two_mp3)*P_mp3,((a-b)%two_mp3)*P_mp3),
+		TuplePoint(((u-2**f)%two_mp3)*Q_mp3,E(0),((a-b)%two_mp3)*Q_mp3,((c-d)%two_mp3)*Q_mp3),
+		TuplePoint(E(0),((u-2**f)%two_mp3)*Q_mp3,((-c-d)%two_mp3)*Q_mp3,((a+b)%two_mp3)*Q_mp3)]
+
+		L_K_dim4=B_K_dim4+[B_K_dim4[0]+B_K_dim4[1]]
+
+		L_K_dim4=[[self._Phi1(TuplePoint(T[0],T[3])),self._Phi2(TuplePoint(T[1],T[2]))] for T in L_K_dim4]
+
+		# Product Theta structure on A_m1*A_m2
+		self.domain_product=ProductThetaStructureDim2To4(self._Phi1._codomain,self._Phi2._codomain)
+		
+		# Theta structure on A_m1*A_m2 after change of theta coordinates
+		self.domain_base_change=self.domain_product.base_change_struct(self.N_dim4)
+		
+		# Converting the kernel to the Theta structure domain_base_change
+		L_K_dim4=[self.domain_product.product_theta_point(T[0],T[1]) for T in L_K_dim4]
+		L_K_dim4=[self.domain_base_change.base_change_coords(self.N_dim4,T) for T in L_K_dim4]
+
+		# Computing the gluing isogeny in dimension 4
+		self._gluing_isogeny_dim4=GluingIsogenyDim4(self.domain_base_change,L_K_dim4,[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),(1,1,0,0)])
+
+		# Translates for the evaluation of the gluing isogeny in dimension 4
+		self.L_trans=[2*B_K_dim4[k] for k in range(2)]
+		self.L_trans_ind=[1,2] # Corresponds to multi indices (1,0,0,0) and (0,1,0,0)
+
+		self._codomain=self._gluing_isogeny_dim4._codomain
+
+	def evaluate(self,P):
+		if not isinstance(P, TuplePoint):
+			raise TypeError("KaniGluingIsogenyChainDim4 isogeny expects as input a TuplePoint on the domain product E^4")
+
+		# Translating P
+		L_P_trans=[P+T for T in self.L_trans]
+		
+		# dim4 --> dim2 x dim2
+		eval_P=[TuplePoint(P[0],P[3]),TuplePoint(P[1],P[2])]
+		eval_L_P_trans=[[TuplePoint(Q[0],Q[3]),TuplePoint(Q[1],Q[2])] for Q in L_P_trans]
+
+		# evaluating through the dimension 2 isogenies
+		eval_P=[self._Phi1(eval_P[0]),self._Phi2(eval_P[1])]
+		eval_L_P_trans=[[self._Phi1(Q[0]),self._Phi2(Q[1])] for Q in eval_L_P_trans]
+
+		# Product Theta structure and base change
+		eval_P=self.domain_product.product_theta_point(eval_P[0],eval_P[1])
+		eval_P=self.domain_base_change.base_change_coords(self.N_dim4,eval_P)
+
+		eval_L_P_trans=[self.domain_product.product_theta_point(Q[0],Q[1]) for Q in eval_L_P_trans]
+		eval_L_P_trans=[self.domain_base_change.base_change_coords(self.N_dim4,Q) for Q in eval_L_P_trans]
+
+		return self._gluing_isogeny_dim4.special_image(eval_P,eval_L_P_trans,self.L_trans_ind)
+
+	def __call__(self,P):
+		return self.evaluate(P)
+
+
+class KaniClapotiGluing:
+	def __init__(self,points_mp3,points_mp2,points_4,integers,strategy_dim2=None,coerce=None):
+		self._coerce=coerce
+		Pu_mp3,Qu_mp3,Pv_mp3,Qv_mp3 = points_mp3
+		Pu_mp2,Qu_mp2,Pv_mp2,Qv_mp2 = points_mp2
+		Pu_4,Qu_4,Pv_4,Qv_4 = points_4
+		gu,xu,yu,gv,xv,yv,Nbk,Nck,e,m = integers
+
+		Eu=Pu_4.curve()
+		Ev=Pv_4.curve()
+
+		lamb_u = inverse_mod(ZZ(gu),4)
+		lamb_v = inverse_mod(ZZ(gv*Nbk*Nck),4)
+
+
+		# 4-torsion canonical change of basis in Eu and Ev (Su=(1,0), Sv=(1,0))
+		_,_,Ru,Su,Mu=make_canonical(Pu_4,lamb_u*Qu_4,4,preserve_pairing=True)
+		_,_,Rv,Sv,Mv=make_canonical(Pv_4,lamb_v*Qv_4,4,preserve_pairing=True)
+
+		Z4 = Integers(4)
+		M0=matrix(Z4,[[Mu[0,0],0,Mu[1,0],0],
+			[0,Mv[0,0],0,Mv[1,0]],
+			[Mu[0,1],0,Mu[1,1],0],
+			[0,Mv[0,1],0,Mv[1,1]]])
+
+		self.M_product_dim2=M0
+
+		# Theta structures in dimension 1 and 2
+		Theta_u=ThetaStructureDim1(Eu,Ru,Su)
+		Theta_v=ThetaStructureDim1(Ev,Rv,Sv)
+
+		Theta_uv=ProductThetaStructureDim2(Theta_u,Theta_v)
+
+		# Gluing change of basis in dimension 2
+		M1 = clapoti_cob_matrix_dim2(integers)
+		M10 = M0*M1
+
+		Fp2 = Eu.base_field()
+		e4 = Fp2(weil_pairing_pari(Ru,Su,4))
+		self.e4 = e4
+
+		N_dim2 = base_change_theta_dim2(M10,e4)
+
+		# Gluing change of basis dimension 2 * dimension 2 --> dimension 4
+		M2 = clapoti_cob_matrix_dim2_dim4(integers)
+
+		self.N_dim4 = base_change_theta_dim4(M2,e4)
+
+		# Kernel of the 2**m-isogeny chain in dimension 2
+		two_mp2=2**(m+2)
+		two_mp3=2*two_mp2
+		u=ZZ(gu*(xu**2+yu**2))
+		mu=inverse_mod(u,two_mp2)
+		suv=ZZ(xu*xv+yu*yv)
+		duv=ZZ(xv*yu-xu*yv)
+		uNbk=(u*Nbk)%two_mp2
+		gusuv=(gu*suv)%two_mp2
+		xK2=(uNbk+gu*gv*mu*Nck*duv**2)%two_mp2
+		B_K_dim2 = [TuplePoint(uNbk*Pu_mp2,gusuv*Pv_mp2),TuplePoint(xK2*Qu_mp2,gusuv*Qv_mp2)]
+
+		# Computation of the 2**m-isogeny chain in dimension 2
+		self._isogenies_dim2=IsogenyChainDim2(B_K_dim2,Theta_uv,N_dim2,m,strategy_dim2)
+
+		# Kernel of the (m+1)-th isogeny in dimension 4 f_{m+1}: A_m^2 --> B (gluing isogeny)
+		xuNbk = (xu*Nbk)%two_mp3
+		yuNbk = (yu*Nbk)%two_mp3
+		inv_Nbk = inverse_mod(Nbk,two_mp3)
+		lambxu = ((1-2**e)*xu)%two_mp3 # extreme case m=e-2, 2^e = 2^(m+2) so 2^e/(u*Nbk) = 2^e mod 2^(m+3).
+		lambyu = ((1-2**e)*yu)%two_mp3
+		xv_Nbk = (xv*inv_Nbk)%two_mp3
+		yv_Nbk = (yv*inv_Nbk)%two_mp3
+
+		B_K_dim4 = [TuplePoint(xuNbk*Pu_mp3,yuNbk*Pu_mp3,xv*Pv_mp3,yv*Pv_mp3),
+		TuplePoint(-yuNbk*Pu_mp3,xuNbk*Pu_mp3,-yv*Pv_mp3,xv*Pv_mp3),
+		TuplePoint(lambxu*Qu_mp3,lambyu*Qu_mp3,xv_Nbk*Qv_mp3,yv_Nbk*Qv_mp3),
+		TuplePoint(-lambyu*Qu_mp3,lambxu*Qu_mp3,-yv_Nbk*Qv_mp3,xv_Nbk*Qv_mp3)]
+
+		L_K_dim4=B_K_dim4+[B_K_dim4[0]+B_K_dim4[1]]
+
+		L_K_dim4=[[self._isogenies_dim2(TuplePoint(L_K_dim4[k][0],L_K_dim4[k][2])),self._isogenies_dim2(TuplePoint(L_K_dim4[k][1],L_K_dim4[k][3]))] for k in range(5)]
+
+		# Product Theta structure on A_m^2
+		self.domain_product=ProductThetaStructureDim2To4(self._isogenies_dim2._codomain,self._isogenies_dim2._codomain)
+		
+		# Theta structure on A_m^2 after change of theta coordinates
+		self.domain_base_change=self.domain_product.base_change_struct(self.N_dim4)
+		
+		# Converting the kernel to the Theta structure domain_base_change
+		L_K_dim4=[self.domain_product.product_theta_point(L_K_dim4[k][0],L_K_dim4[k][1]) for k in range(5)]
+		L_K_dim4=[self.domain_base_change.base_change_coords(self.N_dim4,L_K_dim4[k]) for k in range(5)]
+
+		# Computing the gluing isogeny in dimension 4
+		self._gluing_isogeny_dim4=GluingIsogenyDim4(self.domain_base_change,L_K_dim4,[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),(1,1,0,0)], coerce=self._coerce)
+
+		# Translates for the evaluation of the gluing isogeny in dimension 4
+		self.L_trans=[2*B_K_dim4[k] for k in range(2)]
+		self.L_trans_ind=[1,2] # Corresponds to multi indices (1,0,0,0) and (0,1,0,0)
+
+		self._codomain=self._gluing_isogeny_dim4._codomain
+
+	def evaluate(self,P):
+		if not isinstance(P, TuplePoint):
+			raise TypeError("KaniGluingIsogenyChainDim4 isogeny expects as input a TuplePoint on the domain product Eu^2 x Ev^2")
+
+		# Translating P
+		L_P_trans=[P+T for T in self.L_trans]
+		
+		# dim4 --> dim2 x dim2
+		eval_P=[TuplePoint(P[0],P[2]),TuplePoint(P[1],P[3])]
+		eval_L_P_trans=[[TuplePoint(Q[0],Q[2]),TuplePoint(Q[1],Q[3])] for Q in L_P_trans]
+
+		# evaluating through the dimension 2 isogenies
+		eval_P=[self._isogenies_dim2(eval_P[0]),self._isogenies_dim2(eval_P[1])]
+		eval_L_P_trans=[[self._isogenies_dim2(Q[0]),self._isogenies_dim2(Q[1])] for Q in eval_L_P_trans]
+
+		# Product Theta structure and base change
+		eval_P=self.domain_product.product_theta_point(eval_P[0],eval_P[1])
+		eval_P=self.domain_base_change.base_change_coords(self.N_dim4,eval_P)
+
+		eval_L_P_trans=[self.domain_product.product_theta_point(Q[0],Q[1]) for Q in eval_L_P_trans]
+		eval_L_P_trans=[self.domain_base_change.base_change_coords(self.N_dim4,Q) for Q in eval_L_P_trans]
+
+		return self._gluing_isogeny_dim4.special_image(eval_P,eval_L_P_trans,self.L_trans_ind)
+
+	def __call__(self,P):
+		return self.evaluate(P)
+
+		
+		
+class KaniGluingIsogenyChainDim4:
+	def __init__(self,points_m,points_4,a1,a2,q,m,strategy_dim2=None):
+		r"""
+
+		INPUT: 
+		- points_m: list of 4 points P1_m, Q1_m, R2_m, S2_m of order 2**(m+3)
+		such that (P1_m,Q1_m) generates E1[2**(m+3)] and (R2_m,S2_m) is
+		its image by sigma: E1 --> E2.
+		- points_4: list of 4 points P1_4, Q1_4, R2_4, S2_4 of order 4 obtained by
+		multiplying P1_m, Q1_m, R2_m, S2_m by 2**(m+1).
+		- a1, a2, q: integers such that a1**2+a2**2+q=2**e.
+		- m: 2-adic valuation of a2.
+
+		OUTPUT: Composition of the m+1 first isogenies in the isogeny chained
+		E1^2*E1^2 --> E1^2*E2^2 parametrized by a1, a2 and sigma via Kani's lemma.
+		"""
+
+		P1_m, Q1_m, R2_m, S2_m = points_m
+		P1_4, Q1_4, R2_4, S2_4 = points_4
+
+		E1=P1_m.curve()
+		E2=R2_m.curve()
+
+		Fp2=E1.base_field()
+
+		lamb=inverse_mod(q,4)
+
+		_,_,T1,T2,MT=make_canonical(P1_4,Q1_4,4,preserve_pairing=True)
+		_,_,U1,U2,MU=make_canonical(R2_4,lamb*S2_4,4,preserve_pairing=True)
+
+		Z4=Integers(4)
+		M0=matrix(Z4,[[MT[0,0],0,MT[1,0],0],
+			[0,MU[0,0],0,MU[1,0]],
+			[MT[0,1],0,MT[1,1],0],
+			[0,MU[0,1],0,MU[1,1]]])
+
+		self.M_product_dim2=M0
+
+		# Theta structures in dimension 1 and 2
+		Theta1=ThetaStructureDim1(E1,T1,T2)
+		Theta2=ThetaStructureDim1(E2,U1,U2)
+
+		Theta12=ProductThetaStructureDim2(Theta1,Theta2)
+
+		self.Theta1=Theta1
+		self.Theta2=Theta2
+		self.Theta12=Theta12
+
+		# Gluing base change in dimension 2
+		M1=gluing_base_change_matrix_dim2(a1,a2,q)
+		M10=M0*M1
+
+		self.M_gluing_dim2=M1
+
+		e4=Fp2(weil_pairing_pari(T1,T2,4))
+
+		self.e4=e4
+
+		N_dim2=base_change_theta_dim2(M10,e4)
+		#N_dim2=montgomery_to_theta_matrix_dim2(Theta12.zero().coords(),N1)
+
+		# Gluing base change in dimension 4
+		mua2=-M1[3,1]
+		M2=gluing_base_change_matrix_dim2_dim4(a1,a2,m,mua2)
+
+		self.M_gluing_dim4=M2
+
+		self.N_dim4=base_change_theta_dim4(M2,e4)
+
+		# Kernel of the 2**m-isogeny chain in dimension 2
+		a1_red=a1%(2**(m+2))
+		a2_red=a2%(2**(m+2))
+		B_K_dim2=[TuplePoint(2*a1_red*P1_m-2*a2_red*Q1_m,2*R2_m),TuplePoint(2*a1_red*Q1_m+2*a2_red*P1_m,2*S2_m)]
+
+		# Computation of the 2**m-isogeny chain in dimension 2
+		self._isogenies_dim2=IsogenyChainDim2(B_K_dim2,Theta12,N_dim2,m,strategy_dim2)
+
+		# Kernel of the (m+1)-th isogeny in dimension 4 f_{m+1}: A_m^2 --> B (gluing isogeny)
+		a1_red=a1%(2**(m+3))
+		a2_red=a2%(2**(m+3))
+		
+		a1P1_m=(a1_red)*P1_m
+		a2P1_m=(a2_red)*P1_m
+		a1Q1_m=(a1_red)*Q1_m
+		a2Q1_m=(a2_red)*Q1_m
+
+		OE2=E2(0)
+
+		B_K_dim4=[TuplePoint(a1P1_m,a2P1_m,R2_m,OE2),TuplePoint(a1Q1_m,a2Q1_m,S2_m,OE2),
+				TuplePoint(-a2P1_m,a1P1_m,OE2,R2_m),TuplePoint(-a2Q1_m,a1Q1_m,OE2,S2_m)]
+		L_K_dim4=B_K_dim4+[B_K_dim4[0]+B_K_dim4[1]]
+
+		L_K_dim4=[[self._isogenies_dim2(TuplePoint(L_K_dim4[k][0],L_K_dim4[k][2])),self._isogenies_dim2(TuplePoint(L_K_dim4[k][1],L_K_dim4[k][3]))] for k in range(5)]
+
+		# Product Theta structure on A_m^2
+		self.domain_product=ProductThetaStructureDim2To4(self._isogenies_dim2._codomain,self._isogenies_dim2._codomain)
+		
+		# Theta structure on A_m^2 after base change
+		self.domain_base_change=self.domain_product.base_change_struct(self.N_dim4)
+		
+		# Converting the kernel to the Theta structure domain_base_change
+		L_K_dim4=[self.domain_product.product_theta_point(L_K_dim4[k][0],L_K_dim4[k][1]) for k in range(5)]
+		L_K_dim4=[self.domain_base_change.base_change_coords(self.N_dim4,L_K_dim4[k]) for k in range(5)]
+
+		# Computing the gluing isogeny in dimension 4
+		self._gluing_isogeny_dim4=GluingIsogenyDim4(self.domain_base_change,L_K_dim4,[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),(1,1,0,0)])
+
+		# Translates for the evaluation of the gluing isogeny in dimension 4
+		self.L_trans=[2*B_K_dim4[k] for k in range(2)]
+		self.L_trans_ind=[1,2] # Corresponds to multi indices (1,0,0,0) and (0,1,0,0)
+
+		self._codomain=self._gluing_isogeny_dim4._codomain
+
+	def evaluate(self,P):
+		if not isinstance(P, TuplePoint):
+			raise TypeError("KaniGluingIsogenyChainDim4 isogeny expects as input a TuplePoint on the domain product E1^2 x E2^2")
+
+		# Translating P
+		L_P_trans=[P+T for T in self.L_trans]
+		
+		# dim4 --> dim2 x dim2
+		eval_P=[TuplePoint(P[0],P[2]),TuplePoint(P[1],P[3])]
+		eval_L_P_trans=[[TuplePoint(Q[0],Q[2]),TuplePoint(Q[1],Q[3])] for Q in L_P_trans]
+
+		# evaluating through the dimension 2 isogenies
+		eval_P=[self._isogenies_dim2(eval_P[0]),self._isogenies_dim2(eval_P[1])]
+		eval_L_P_trans=[[self._isogenies_dim2(Q[0]),self._isogenies_dim2(Q[1])] for Q in eval_L_P_trans]
+
+		# Product Theta structure and base change
+		eval_P=self.domain_product.product_theta_point(eval_P[0],eval_P[1])
+		eval_P=self.domain_base_change.base_change_coords(self.N_dim4,eval_P)
+
+		eval_L_P_trans=[self.domain_product.product_theta_point(Q[0],Q[1]) for Q in eval_L_P_trans]
+		eval_L_P_trans=[self.domain_base_change.base_change_coords(self.N_dim4,Q) for Q in eval_L_P_trans]
+
+		return self._gluing_isogeny_dim4.special_image(eval_P,eval_L_P_trans,self.L_trans_ind)
+
+	def __call__(self,P):
+		return self.evaluate(P)
+
+class KaniGluingIsogenyChainDim4Half:
+	def __init__(self, points_m, a1, a2, q, m, Theta12, M_product_dim2, M_start_dim4, M_gluing_dim4, e4, dual=False,strategy_dim2=None):#points_m,points_4,a1,a2,q,m,precomputed_data=None,dual=False,strategy_dim2=None):
+		r"""
+
+		INPUT: 
+		- points_m: list of 4 points P1_m, Q1_m, R2_m, S2_m of order 2**(m+3)
+		such that (P1_m,Q1_m) generates E1[2**(m+3)] and (R2_m,S2_m) is
+		its image by sigma: E1 --> E2.
+		- points_4: list of 4 points P1_4, Q1_4, R2_4, S2_4 of order 4 obtained by
+		multiplying P1_m, Q1_m, R2_m, S2_m by 2**(m+1).
+		- a1, a2, q: integers such that a1**2+a2**2+q=2**e.
+		- m: 2-adic valuation of a2.
+
+		OUTPUT: Composition of the m+1 first isogenies in the isogeny chained
+		E1^2*E1^2 --> E1^2*E2^2 parametrized by a1, a2 and sigma via Kani's lemma.
+		"""
+
+		P1_m, Q1_m, R2_m, S2_m = points_m
+
+		E1=P1_m.curve()
+		E2=R2_m.curve()
+
+		Fp2=E1.base_field()
+
+		self.M_product_dim2 = M_product_dim2
+
+		self.Theta12=Theta12
+
+		self.e4=e4
+
+		# Gluing base change in dimension 2
+		if not dual:
+			M1=gluing_base_change_matrix_dim2_F1(a1,a2,q)
+		else:
+			M1=gluing_base_change_matrix_dim2_F2(a1,a2,q)
+		
+		M10=M_product_dim2*M1
+
+		self.M_gluing_dim2=M1
+
+		self.e4=e4
+
+		N_dim2=base_change_theta_dim2(M10,e4)
+		#N_dim2=montgomery_to_theta_matrix_dim2(Theta12.zero().coords(),N1)
+
+		# Gluing base change in dimension 4
+
+		self.M_gluing_dim4 = M_gluing_dim4
+
+		self.N_dim4 = base_change_theta_dim4(M_gluing_dim4, e4)
+
+		# Kernel of the 2**m-isogeny chain in dimension 2
+		a1_red=a1%(2**(m+2))
+		a2_red=a2%(2**(m+2))
+		if not dual:
+			B_K_dim2=[TuplePoint(2*a1_red*P1_m-2*a2_red*Q1_m,2*R2_m),TuplePoint(2*a1_red*Q1_m+2*a2_red*P1_m,2*S2_m)]
+		else:
+			B_K_dim2=[TuplePoint(2*a1_red*P1_m+2*a2_red*Q1_m,-2*R2_m),TuplePoint(2*a1_red*Q1_m-2*a2_red*P1_m,-2*S2_m)]
+
+		# Computation of the 2**m-isogeny chain in dimension 2
+		self._isogenies_dim2=IsogenyChainDim2(B_K_dim2,Theta12,N_dim2,m,strategy_dim2)
+
+		# Kernel of the (m+1)-th isogeny in dimension 4 f_{m+1}: A_m^2 --> B (gluing isogeny)
+		lamb=inverse_mod(q,2**(m+3))
+		B_K_dim4=kernel_basis(M_start_dim4,m+1,P1_m,Q1_m,R2_m,lamb*S2_m)
+		L_K_dim4=B_K_dim4+[B_K_dim4[0]+B_K_dim4[1]]
+
+		L_K_dim4=[[self._isogenies_dim2(TuplePoint(L_K_dim4[k][0],L_K_dim4[k][2])),self._isogenies_dim2(TuplePoint(L_K_dim4[k][1],L_K_dim4[k][3]))] for k in range(5)]
+
+		# Product Theta structure on A_m^2
+		self.domain_product=ProductThetaStructureDim2To4(self._isogenies_dim2._codomain,self._isogenies_dim2._codomain)
+		
+		# Theta structure on A_m^2 after base change
+		self.domain_base_change=self.domain_product.base_change_struct(self.N_dim4)
+		
+		# Converting the kernel to the Theta structure domain_base_change
+		L_K_dim4=[self.domain_product.product_theta_point(L_K_dim4[k][0],L_K_dim4[k][1]) for k in range(5)]
+		L_K_dim4=[self.domain_base_change.base_change_coords(self.N_dim4,L_K_dim4[k]) for k in range(5)]
+
+		# Computing the gluing isogeny in dimension 4
+		self._gluing_isogeny_dim4=GluingIsogenyDim4(self.domain_base_change,L_K_dim4,[(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),(1,1,0,0)])
+
+		# Translates for the evaluation of the gluing isogeny in dimension 4
+		self.L_trans=[2*B_K_dim4[k] for k in range(2)]
+		self.L_trans_ind=[1,2] # Corresponds to multi indices (1,0,0,0) and (0,1,0,0)
+
+		self._codomain=self._gluing_isogeny_dim4._codomain
+
+	def evaluate(self,P):
+		if not isinstance(P, TuplePoint):
+			raise TypeError("KaniGluingIsogenyChainDim4 isogeny expects as input a TuplePoint on the domain product E1^2 x E2^2")
+
+		# Translating P
+		L_P_trans=[P+T for T in self.L_trans]
+		
+		# dim4 --> dim2 x dim2
+		eval_P=[TuplePoint(P[0],P[2]),TuplePoint(P[1],P[3])]
+		eval_L_P_trans=[[TuplePoint(Q[0],Q[2]),TuplePoint(Q[1],Q[3])] for Q in L_P_trans]
+
+		# evaluating through the dimension 2 isogenies
+		eval_P=[self._isogenies_dim2(eval_P[0]),self._isogenies_dim2(eval_P[1])]
+		eval_L_P_trans=[[self._isogenies_dim2(Q[0]),self._isogenies_dim2(Q[1])] for Q in eval_L_P_trans]
+
+		# Product Theta structure and base change
+		eval_P=self.domain_product.product_theta_point(eval_P[0],eval_P[1])
+		eval_P=self.domain_base_change.base_change_coords(self.N_dim4,eval_P)
+
+		eval_L_P_trans=[self.domain_product.product_theta_point(Q[0],Q[1]) for Q in eval_L_P_trans]
+		eval_L_P_trans=[self.domain_base_change.base_change_coords(self.N_dim4,Q) for Q in eval_L_P_trans]
+
+		return self._gluing_isogeny_dim4.special_image(eval_P,eval_L_P_trans,self.L_trans_ind)
+
+	def __call__(self,P):
+		return self.evaluate(P)
+
+	def dual(self):
+		domain = self._codomain.hadamard()
+		codomain_base_change = self.domain_base_change
+		codomain_product = self.domain_product
+		N_dim4 = self.N_dim4.inverse()
+		isogenies_dim2 = self._isogenies_dim2.dual()
+		splitting_isogeny_dim4 = self._gluing_isogeny_dim4.dual()
+
+		return KaniSplittingIsogenyChainDim4(domain, codomain_base_change, codomain_product, N_dim4, isogenies_dim2, splitting_isogeny_dim4)
+
+class KaniSplittingIsogenyChainDim4:
+	def __init__(self, domain, codomain_base_change, codomain_product, N_dim4, isogenies_dim2, splitting_isogeny_dim4):
+		self._domain = domain
+		self.codomain_base_change = codomain_base_change
+		self.codomain_product = codomain_product
+		self.N_dim4 = N_dim4
+		self._isogenies_dim2 = isogenies_dim2
+		self._splitting_isogeny_dim4 = splitting_isogeny_dim4
+
+	def evaluate(self,P):
+		if not isinstance(P, ThetaPointDim4):
+			raise TypeError("KaniSplittingIsogenyChainDim4 isogeny expects as input a ThetaPointDim4")
+
+		Q = self._splitting_isogeny_dim4(P)
+		Q = self.codomain_product.base_change_coords(self.N_dim4, Q)
+		Q1, Q2 = self.codomain_product.to_theta_points(Q)
+		Q1, Q2 = self._isogenies_dim2._domain(Q1.hadamard()), self._isogenies_dim2._domain(Q2.hadamard()) 
+
+		Q1 = self._isogenies_dim2(Q1)
+		Q2 = self._isogenies_dim2(Q2)
+
+		return TuplePoint(Q1[0],Q2[0],Q1[1],Q2[1])
+
+	def __call__(self,P):
+		return self.evaluate(P)