From d40de259097c5e8d8fd35539560ca7c3d47523e7 Mon Sep 17 00:00:00 2001
From: Ryan Rueger <git@rueg.re>
Date: Sat, 1 Mar 2025 20:25:41 +0100
Subject: Initial Commit
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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>
---
 theta_lib/theta_structures/Theta_dim4.py | 351 +++++++++++++++++++++++++++++++
 1 file changed, 351 insertions(+)
 create mode 100644 theta_lib/theta_structures/Theta_dim4.py

(limited to 'theta_lib/theta_structures/Theta_dim4.py')

diff --git a/theta_lib/theta_structures/Theta_dim4.py b/theta_lib/theta_structures/Theta_dim4.py
new file mode 100644
index 0000000..7851c56
--- /dev/null
+++ b/theta_lib/theta_structures/Theta_dim4.py
@@ -0,0 +1,351 @@
+from sage.all import *
+from sage.structure.element import get_coercion_model
+
+from .theta_helpers_dim4 import (
+    hadamard,
+    batch_inversion,
+    product_theta_point_dim4,
+    product_to_theta_points_dim4,
+    product_theta_point_dim2_dim4,
+    product_to_theta_points_dim4_dim2,
+    act_point,
+    squared,
+)
+from .Theta_dim1 import ThetaStructureDim1
+from .Tuple_point import TuplePoint
+from ..basis_change.base_change_dim4 import (
+    apply_base_change_theta_dim4,
+    random_symplectic_matrix,
+    base_change_theta_dim4,
+)
+
+cm = get_coercion_model()
+
+
+class ThetaStructureDim4:
+	def __init__(self,null_point,null_point_dual=None,inv_null_point_dual_sq=None):
+		r"""
+		INPUT:
+		- null_point: theta-constants.
+		- inv_null_point_dual_sq: inverse of the squares of dual theta-constants, if provided 
+		(meant to prevent duplicate computation, since this data is already computed when the 
+		codomain of an isogeny is computed).
+		"""
+		if not len(null_point) == 16:
+			raise ValueError("Entry null_point should have 16 coordinates.")
+
+		self._base_ring = cm.common_parent(*(c.parent() for c in null_point))
+		self._point = ThetaPointDim4
+		self._null_point = self._point(self, null_point)
+		self._null_point_dual=null_point_dual
+		self._inv_null_point=None
+		self._inv_null_point_dual_sq=inv_null_point_dual_sq
+
+	def null_point(self):
+		"""
+		"""
+		return self._null_point
+
+	def null_point_dual(self):
+		if self._null_point_dual==None:
+			self._null_point_dual=hadamard(self._null_point.coords())
+		return self._null_point_dual
+
+	def base_ring(self):
+		"""
+		"""
+		return self._base_ring
+
+	def zero(self):
+		"""
+		"""
+		return self.null_point()
+
+	def zero_dual(self):
+		return self.null_point_dual()
+
+	def __repr__(self):
+		return f"Theta structure over {self.base_ring()} with null point: {self.null_point()}"
+
+	def __call__(self,coords):
+		return self._point(self,coords)
+
+	def act_null(self,I,J):
+		r"""
+		Point of 2-torsion.
+
+		INPUT:
+		- I, J: two 4-tuples of indices in {0,1}.
+
+		OUTPUT: the action of (I,\chi_J) on the theta null point given by:
+		(I,\chi_J)*P=(\chi_J(I+K)^{-1}P[I+K])_K
+		"""
+		return self.null_point().act_point(I,J)
+
+	def is_K2(self,B):
+		r"""
+		Given a symplectic decomposition A[2]=K_1\oplus K_2 canonically 
+		induced by the theta-null point, determines if B is the canonical
+		basis of K_2 given by act_nul(0,\delta_i)_{1\leq i\leq 4}.
+
+		INPUT:
+		- B: Basis of 4 points of 2-torsion.
+
+		OUTPUT: Boolean True if and only if B is the canonical basis of K_2.
+		"""
+		I0=(0,0,0,0)
+		if B[0]!=self.act_null(I0,(1,0,0,0)):
+			return False
+		if B[1]!=self.act_null(I0,(0,1,0,0)):
+			return False
+		if B[2]!=self.act_null(I0,(0,0,1,0)):
+			return False
+		if B[3]!=self.act_null(I0,(0,0,0,1)):
+			return False
+		return True
+
+	def base_change_struct(self,N):
+		null_coords=self.null_point().coords()
+		new_null_coords=apply_base_change_theta_dim4(N,null_coords)
+		return ThetaStructureDim4(new_null_coords)
+
+	def base_change_coords(self,N,P):
+		coords=P.coords()
+		new_coords=apply_base_change_theta_dim4(N,coords)
+		return self.__call__(new_coords)
+
+	#@cached_method
+	def _arithmetic_precomputation(self):
+		r"""
+		Initializes the precomputation containing the inverse of the theta-constants in standard
+		and dual (Hadamard transformed) theta-coordinates. Assumes no theta-constant is zero.
+		"""
+		O=self.null_point()
+		if all([O[k]!=0 for k in range(16)]):
+			self._inv_null_point=batch_inversion(O.coords())
+			if self._inv_null_point_dual_sq==None:
+				U_chi_0_sq=hadamard(squared(O.coords()))
+				if all([U_chi_0_sq[k]!=0 for k in range(16)]):	
+					self._inv_null_point_dual_sq=batch_inversion(U_chi_0_sq)
+					self._arith_base_change=False
+				else:
+					self._arith_base_change=True
+					self._arithmetic_base_change()
+					#print("Zero dual theta constants.\nRandom symplectic base change is being used for duplication.\nDoublings are more costly than expected.")
+		else:
+			self._arith_base_change=True
+			self._arithmetic_base_change()
+			#print("Zero theta constants.\nRandom symplectic base change is being used for duplication.\nDoublings are more costly than expected.")
+
+	def _arithmetic_base_change(self,max_iter=50):
+		F=self._base_ring
+		if F.degree() == 2:
+			i=self._base_ring.gen()
+		else:
+			assert(F.degree() == 1)
+			Fp2 = GF((F.characteristic(), 2), name='i', modulus=var('x')**2 + 1)
+			i=Fp2.gen()
+
+		count=0
+		O=self.null_point()
+		while count<max_iter:
+			count+=1
+			M=random_symplectic_matrix(4)
+			N=base_change_theta_dim4(M,i)
+
+			NO=apply_base_change_theta_dim4(N,O)
+			NU_chi_0_sq=hadamard(squared(NO))
+
+			if all([NU_chi_0_sq[k]!=0 for k in range(16)]) and all([NO[k]!=0 for k in range(16)]):
+				self._arith_base_change_matrix=N
+				self._arith_base_change_matrix_inv=N.inverse()
+				self._inv_null_point=batch_inversion(NO)
+				self._inv_null_point_dual_sq=batch_inversion(NU_chi_0_sq)
+				break
+
+	def has_suitable_doubling(self):
+		O=self.null_point()
+		UO=hadamard(O.coords())
+		if all([O[k]!=0 for k in range(16)]) and all([UO[k]!=0 for k in range(16)]):
+			return True
+		else:
+			return False
+
+	def hadamard(self):
+		return ThetaStructureDim4(self.null_point_dual())
+
+	def hadamard_change_coords(self,P):
+		new_coords=hadamard(P)
+		return self.__call__(new_coords)
+
+
+class ProductThetaStructureDim1To4(ThetaStructureDim4):
+	def __init__(self,*args):
+		r"""Defines the product theta structure at level 2 of 4 elliptic curves.
+
+		Input: Either
+		- 4 theta structures of dimension 1: T0, T1, T2, T3;
+		- 4 elliptic curves: E0, E1, E2, E3.
+		- 4 elliptic curves E0, E1, E2, E3 and their respective canonical 4-torsion basis B0, B1, B2, B3.
+		"""
+		if len(args)==4:
+			theta_structures=list(args)
+			for k in range(4):
+				if not isinstance(theta_structures[k],ThetaStructureDim1):
+					try:
+						theta_structures[k]=ThetaStructureDim1(theta_structures[k])
+					except:
+						pass
+		elif len(args)==8:
+			theta_structures=[ThetaStructureDim1(args[k],args[4+k][0],args[4+k][1]) for k in range(4)]
+		else:
+			raise ValueError("4 or 8 arguments expected but {} were given.\nYou should enter a list of 4 elliptic curves or ThetaStructureDim1\nor a list of 4 elliptic curves with a 4-torsion basis for each of them.".format(len(args)))
+
+		self._theta_structures=theta_structures
+
+		null_point=product_theta_point_dim4(theta_structures[0].zero().coords(),theta_structures[1].zero().coords(),
+			theta_structures[2].zero().coords(),theta_structures[3].zero().coords())
+
+		ThetaStructureDim4.__init__(self,null_point)
+
+	def product_theta_point(self,theta_points):
+		return self._point(self,product_theta_point_dim4(theta_points[0].coords(),theta_points[1].coords(),
+			theta_points[2].coords(),theta_points[3].coords()))
+
+	def __call__(self,point):
+		if isinstance(point,TuplePoint):
+			theta_points=[]
+			theta_structures=self._theta_structures
+			for i in range(4):
+				theta_points.append(theta_structures[i](point[i]))
+			return self.product_theta_point(theta_points)
+		else:
+			return self._point(self,point)
+
+	def to_theta_points(self,P):
+		theta_coords=product_to_theta_points_dim4(P)
+		theta_points=[self._theta_structures[i](theta_coords[i]) for i in range(4)]
+		return theta_points
+
+	def to_tuple_point(self,P):
+		theta_points=self.to_theta_points(P)
+		montgomery_points=[self._theta_structures[i].to_montgomery_point(theta_points[i]) for i in range(4)]
+		return TuplePoint(montgomery_points)
+
+class ProductThetaStructureDim2To4(ThetaStructureDim4):
+	def __init__(self,theta1,theta2):
+		self._theta_structures=(theta1,theta2)
+
+		null_point=product_theta_point_dim2_dim4(theta1.zero().coords(),theta2.zero().coords())
+
+		ThetaStructureDim4.__init__(self,null_point)
+
+	def product_theta_point(self,P1,P2):
+		return self._point(self,product_theta_point_dim2_dim4(P1.coords(),P2.coords()))
+
+	def __call__(self,point):
+		return self._point(self,point)
+
+	def to_theta_points(self,P):
+		theta_coords=product_to_theta_points_dim4_dim2(P)
+		theta_points=[self._theta_structures[i](theta_coords[i]) for i in range(2)]
+		return theta_points
+
+class ThetaPointDim4:
+    def __init__(self, parent, coords):
+        """
+        """
+        if not isinstance(parent, ThetaStructureDim4):
+            raise ValueError("Entry parent should be a ThetaStructureDim4 object.")
+
+        self._parent = parent
+        self._coords = tuple(coords)
+
+    def parent(self):
+        """
+        """
+        return self._parent
+    
+    def theta(self):
+        """
+        """
+        return self.parent()
+    
+    def coords(self):
+        """
+        """
+        return self._coords
+    
+    def is_zero(self):
+        """
+        """
+        return self == self.parent().zero()
+
+    def __eq__(self, other):
+    	P=self.coords()
+    	Q=other.coords()
+    	
+    	k0=0
+    	while k0<15 and P[k0]==0:
+    		k0+=1
+
+    	for l in range(16):
+    		if P[l]*Q[k0]!=Q[l]*P[k0]:
+    			return False
+    	return True
+
+    def __repr__(self):
+    	return f"Theta point with coordinates: {self.coords()}"
+
+    def __getitem__(self,i):
+    	return self._coords[i]
+
+    def scale(self,lamb):
+    	if lamb==0:
+    		raise ValueError("Entry lamb should be non-zero.")
+
+    	P=self.coords()
+    	return self._parent([lamb*x for x in P])
+
+    def act_point(self,I,J):
+    	r"""
+		Translation by a point of 2-torsion.
+
+		Input:
+		- I, J: two 4-tuples of indices in {0,1}.
+
+		Output: the action of (I,\chi_J) on P given by:
+		(I,\chi_J)*P=(\chi_J(I+K)^{-1}P[I+K])_K
+		"""
+    	return self._parent(act_point(self._coords,I,J))
+
+    def double(self):
+    	## This formula is projective.
+    	## Works only when theta constants are non-zero.
+    	P=self.coords()
+    	if self.parent()._inv_null_point==None or self.parent()._inv_null_point_dual_sq==None:
+    		self.parent()._arithmetic_precomputation()
+    	inv_O,inv_U_chi_0_sq=self.parent()._inv_null_point,self.parent()._inv_null_point_dual_sq
+
+    	if self.parent()._arith_base_change:
+    		P=apply_base_change_theta_dim4(self.parent()._arith_base_change_matrix,P)
+
+    	U_chi_P=squared(hadamard(squared(P)))
+    	for chi in range(16):
+    		U_chi_P[chi]*=inv_U_chi_0_sq[chi]
+
+    	theta_2P = list(hadamard(U_chi_P))
+    	for i in range(16):
+    		theta_2P[i] *= inv_O[i]
+
+    	if self.parent()._arith_base_change:
+    		theta_2P=apply_base_change_theta_dim4(self.parent()._arith_base_change_matrix_inv,theta_2P)
+
+    	return self._parent(theta_2P)
+
+    def double_iter(self,n):
+    	## Computes 2**n*self
+    	Q=self
+    	for i in range(n):
+    		Q=Q.double()
+    	return Q
-- 
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