From d40de259097c5e8d8fd35539560ca7c3d47523e7 Mon Sep 17 00:00:00 2001 From: Ryan Rueger Date: Sat, 1 Mar 2025 20:25:41 +0100 Subject: Initial Commit MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-Authored-By: Damien Robert Co-Authored-By: Frederik Vercauteren Co-Authored-By: Jonathan Komada Eriksen Co-Authored-By: Pierrick Dartois Co-Authored-By: Riccardo Invernizzi Co-Authored-By: Ryan Rueger [0.01s] Co-Authored-By: Benjamin Wesolowski Co-Authored-By: Arthur Herlédan Le Merdy Co-Authored-By: Boris Fouotsa --- theta_lib/basis_change/kani_base_change.py | 975 +++++++++++++++++++++++++++++ 1 file changed, 975 insertions(+) create mode 100644 theta_lib/basis_change/kani_base_change.py (limited to 'theta_lib/basis_change/kani_base_change.py') diff --git a/theta_lib/basis_change/kani_base_change.py b/theta_lib/basis_change/kani_base_change.py new file mode 100644 index 0000000..e6de2e2 --- /dev/null +++ b/theta_lib/basis_change/kani_base_change.py @@ -0,0 +1,975 @@ +from sage.all import * +from ..basis_change.canonical_basis_dim1 import make_canonical +from ..basis_change.base_change_dim2 import is_symplectic_matrix_dim2 +from ..basis_change.base_change_dim4 import ( + complete_symplectic_matrix_dim4, + is_symplectic_matrix_dim4, + bloc_decomposition, +) +from ..theta_structures.Tuple_point import TuplePoint + + +def base_change_canonical_dim2(P1,P2,R1,R2,q,f): + r""" + + Input: + - P1, P2: basis of E1[2**f]. + - R1, R2: images of P1, P2 by \sigma: E1 --> E2. + - q: degree of \sigma. + - f: log_2(order of P1 and P2). + + Output: + - P1_doubles: list of 2**i*P1 for i in {0,...,f-2}. + - P2_doubles: list of 2**i*P2 for i in {0,...,f-2}. + - R1_doubles: list of 2**i*R1 for i in {0,...,f-2}. + - R2_doubles: list of 2**i*R2 for i in {0,...,f-2}, + - T1, T2: canonical basis of E1[4]. + - U1, U2: canonical basis of E2[4]. + - M0: base change matrix of the symplectic basis 2**(f-2)*B1 of E1*E2[4] given by: + B1:=[[(P1,0),(0,R1)],[(P2,0),(0,lamb*R2)]] + where lamb is the modular inverse of q mod 2**f, so that: + e_{2**f}(P1,P2)=e_{2**f}(R1,lamb*R2). + in the canonical symplectic basis: + B0:=[[(T1,0),(0,U1)],[(T2,0),(0,U2)]]. + """ + lamb=inverse_mod(q,4) + + P1_doubles=[P1] + P2_doubles=[P2] + R1_doubles=[R1] + R2_doubles=[R2] + + for i in range(f-2): + P1_doubles.append(2*P1_doubles[-1]) + P2_doubles.append(2*P2_doubles[-1]) + R1_doubles.append(2*R1_doubles[-1]) + R2_doubles.append(2*R2_doubles[-1]) + + # Constructing canonical basis of E1[4] and E2[4]. + _,_,T1,T2,MT=make_canonical(P1_doubles[-1],P2_doubles[-1],4,preserve_pairing=True) + _,_,U1,U2,MU=make_canonical(R1_doubles[-1],lamb*R2_doubles[-1],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]]]) + + return P1_doubles,P2_doubles,R1_doubles,R2_doubles,T1,T2,U1,U2,M0 + +def gluing_base_change_matrix_dim2(a1,a2,q): + r"""Computes the symplectic base change matrix of a symplectic basis (*,B_K4) of E1*E2[4] + given by the kernel of the dimension 2 gluing isogeny: + B_K4=2**(f-2)[([a1]P1-[a2]P2,R1),([a1]P2+[a2]P1,R2)] + in the basis $2**(f-2)*B1$ given by: + B1:=[[(P1,0),(0,R1)],[(P2,0),(0,lamb*R2)]] + where: + - lamb is the inverse of q modulo 2**f. + - (P1,P2) is the canonical basis of E1[2**f]. + - (R1,R2) is the image of (P1,P2) by sigma. + + Input: + - a1, q: integers. + + Output: + - M: symplectic base change matrix of (*,B_K4) in 2**(f-2)*B1. + """ + + Z4=Integers(4) + + mu=inverse_mod(a1,4) + + A=matrix(Z4,[[0,mu], + [0,0]]) + B=matrix(Z4,[[0,0], + [-1,-ZZ(mu*a2)]]) + + C=matrix(Z4,[[ZZ(a1),ZZ(a2)], + [1,0]]) + D=matrix(Z4,[[-ZZ(a2),ZZ(a1)], + [0,ZZ(q)]]) + + #M=complete_symplectic_matrix_dim2(C, D, 4) + M=block_matrix([[A,C],[B,D]]) + + assert is_symplectic_matrix_dim2(M) + + return M + +# ============================================== # +# Functions for the class KaniClapotiIsog # +# ============================================== # + +def clapoti_cob_matrix_dim2(integers): + gu,xu,yu,gv,xv,yv,Nbk,Nck,e,m = integers + + xu = ZZ(xu) + xv = ZZ(xv) + Nbk = ZZ(Nbk) + Nck = ZZ(Nck) + u = ZZ(gu*(xu**2+yu**2)) + v = ZZ(gv*(xv**2+yv**2)) + mu = inverse_mod(u,4) + suv = xu*xv+yu*yv + inv_Nbk = inverse_mod(Nbk,4) + inv_gugvNcksuv = inverse_mod(gu*gv*Nck*suv,4) + + Z4=Integers(4) + + M=matrix(Z4,[[0,0,u*Nbk,0], + [0,inv_Nbk*inv_gugvNcksuv,gu*suv,0], + [-inv_Nbk*mu,0,0,gu*Nbk*u], + [0,0,0,gu*gv*Nbk*Nck*suv]]) + + assert is_symplectic_matrix_dim2(M) + + return M + +def clapoti_cob_matrix_dim2_dim4(integers): + gu,xu,yu,gv,xv,yv,Nbk,Nck,e,m = integers + + xu = ZZ(xu) + yu = ZZ(yu) + xv = ZZ(xv) + yv = ZZ(yv) + gu = ZZ(gu) + gv = ZZ(gv) + Nbk = ZZ(Nbk) + Nck = ZZ(Nck) + u = ZZ(gu*(xu**2+yu**2)) + v = ZZ(gv*(xv**2+yv**2)) + suv = xu*xv+yu*yv + duv = xv*yu-xu*yv + duv_2m = duv//2**m + mu = inverse_mod(u,4) + nu = inverse_mod(v,4) + sigmauv = inverse_mod(suv,4) + inv_guNbk = inverse_mod(gu*Nbk,4) + lamb = nu*gu*gv*Nbk*suv + mu1 = ZZ(mu*gu**2*gv*suv*Nbk*Nck*duv_2m) + mu2 = ZZ(duv_2m*gu*sigmauv*(Nbk*u*yu+gv*xv*Nck*duv)) + mu3 = ZZ(duv_2m*gu*sigmauv*(Nbk*u*xu-gv*yv*Nck*duv)) + + Z4=Integers(4) + + M=matrix(Z4,[[gu*xu,-gu*yu,0,0,0,0,mu2,mu3], + [0,0,lamb*xv,-lamb*yv,mu1*yu,mu1*xu,0,0], + [gu*yu,gu*xu,0,0,0,0,-mu3,mu2], + [0,0,lamb*yv,lamb*xv,-mu1*xu,mu1*yu,0,0], + [0,0,0,0,mu*xu,-mu*yu,0,0], + [0,0,0,0,0,0,inv_guNbk*xv*sigmauv,-inv_guNbk*yv*sigmauv], + [0,0,0,0,mu*yu,mu*xu,0,0], + [0,0,0,0,0,0,inv_guNbk*yv*sigmauv,inv_guNbk*xv*sigmauv]]) + + assert is_symplectic_matrix_dim4(M) + + return M + +def clapoti_cob_splitting_matrix(integers): + gu,xu,yu,gv,xv,yv,Nbk,Nck,e,m = integers + + v=ZZ(gv*(xv**2+yv**2)) + vNck=ZZ(v*Nck) + inv_vNck=inverse_mod(vNck,4) + + Z4=Integers(4) + + M=matrix(Z4,[[0,0,0,0,-1,0,0,0], + [0,0,0,0,0,-1,0,0], + [0,0,vNck,0,0,0,0,0], + [0,0,0,vNck,0,0,0,0], + [1,0,-vNck,0,0,0,0,0], + [0,1,0,-vNck,0,0,0,0], + [0,0,0,0,1,0,inv_vNck,0], + [0,0,0,0,0,1,0,inv_vNck]]) + + assert is_symplectic_matrix_dim4(M) + + return M + +# =============================================== # +# Functions for the class KaniFixedDegDim2 # +# =============================================== # + +def fixed_deg_gluing_matrix_Phi1(u,a,b,c,d): + u,a,b,c,d = ZZ(u),ZZ(a),ZZ(b),ZZ(c),ZZ(d) + + mu = inverse_mod(u,4) + inv_cmd = inverse_mod(c-d,4) + + Z4 = Integers(4) + + M = matrix(Z4,[[0,0,u,0], + [0,inv_cmd,c+d,0], + [-mu,0,0,(d**2-c**2)*mu], + [0,0,0,c-d]]) + + assert is_symplectic_matrix_dim2(M) + + return M + +def fixed_deg_gluing_matrix_Phi2(u,a,b,c,d): + u,a,b,c,d = ZZ(u),ZZ(a),ZZ(b),ZZ(c),ZZ(d) + + mu = inverse_mod(u,4) + inv_cpd = inverse_mod(c+d,4) + + Z4 = Integers(4) + + M = matrix(Z4,[[0,0,u,0], + [0,-inv_cpd,d-c,0], + [-mu,0,0,(d**2-c**2)*mu], + [0,0,0,-(c+d)]]) + + assert is_symplectic_matrix_dim2(M) + + return M + +def fixed_deg_gluing_matrix_dim4(u,a,b,c,d,m): + u,a,b,c,d = ZZ(u),ZZ(a),ZZ(b),ZZ(c),ZZ(d) + + mu = inverse_mod(u,4) + nu = ZZ((-mu**2)%4) + amb_2m = ZZ((a-b)//2**m) + apb_2m = ZZ((a+b)//2**m) + u2pc2md2_2m = ZZ((u**2+c**2-d**2)//2**m) + inv_cmd = inverse_mod(c-d,4) + inv_cpd = inverse_mod(c+d,4) + + + Z4 = Integers(4) + + M = matrix(Z4,[[1,0,0,0,0,0,-u2pc2md2_2m,-apb_2m*(c+d)], + [0,0,(c+d)*(c-d)*nu,(a-b)*(c-d)*nu,0,amb_2m*(c-d),0,0], + [0,1,0,0,0,0,amb_2m*(c-d),-u2pc2md2_2m], + [0,0,-(a+b)*(c+d)*nu,(c+d)*(c-d)*nu,-apb_2m*(c+d),0,0,0], + [0,0,0,0,1,0,0,0], + [0,0,0,0,0,0,1,(a+b)*inv_cmd], + [0,0,0,0,0,1,0,0], + [0,0,0,0,0,0,(b-a)*inv_cpd,1]]) + + assert is_symplectic_matrix_dim4(M) + + return M + +def fixed_deg_gluing_matrix(u,a,b,c,d): + r""" + Deprecated. + """ + + mu = inverse_mod(u,4) + nu = (-mu**2)%4 + + Z4 = Integers(4) + + M = matrix(Z4,[[0,0,0,0,ZZ(u),0,0,0], + [0,0,0,0,0,ZZ(u),0,0], + [0,0,ZZ(nu*(a+b)),ZZ(nu*(d-c)),ZZ(a+b),ZZ(d-c),0,0], + [0,0,ZZ(nu*(c+d)),ZZ(nu*(a-b)),ZZ(c+d),ZZ(a-b),0,0], + [ZZ(-mu),0,0,0,0,0,ZZ(u),0], + [0,ZZ(-mu),0,0,0,0,0,ZZ(u)], + [0,0,0,0,0,0,ZZ(a-b),ZZ(-c-d)], + [0,0,0,0,0,0,ZZ(c-d),ZZ(a+b)]]) + + assert is_symplectic_matrix_dim4(M) + + return M + +def fixed_deg_splitting_matrix(u): + + mu = inverse_mod(u,4) + + Z4 = Integers(4) + + M = matrix(Z4,[[0,0,0,0,-1,0,0,0], + [0,0,0,0,0,-1,0,0], + [0,0,ZZ(-u),0,0,0,0,0], + [0,0,0,ZZ(-u),0,0,0,0], + [1,0,ZZ(-mu),0,0,0,0,0], + [0,1,0,ZZ(-mu),0,0,0,0], + [0,0,0,0,ZZ(mu),0,ZZ(mu),0], + [0,0,0,0,0,ZZ(mu),0,ZZ(mu)]]) + + assert is_symplectic_matrix_dim4(M) + + return M + + +# ========================================================== # +# Functions for the class KaniEndo (one isogeny chain) # +# ========================================================== # + +def gluing_base_change_matrix_dim2_dim4(a1,a2,m,mua2): + r"""Computes the symplectic base change matrix of a symplectic basis (*,B_K4) of Am*Am[4] + given by the kernel of the dimension 4 gluing isogeny Am*Am-->B: + + B_K4=2**(e-m)[(Phi([a1]P1,sigma(P1)),Phi([a2]P1,0)),(Phi([a1]P2,sigma(P2)),Phi([a2]P2,0)), + (Phi(-[a2]P1,0),Phi([a1]P1,sigma(P1))),(Phi(-[a2]P2,0),Phi([a2]P2,sigma(P2)))] + + in the basis associated to the product theta-structure of level 2 of Am*Am: + + B:=[(S1,0),(S2,0),(0,S1),(0,S2),(T1,0),(T2,0),(0,T1),(0,T2)] + + where: + - (P1,P2) is the canonical basis of E1[2**f]. + - (R1,R2) is the image of (P1,P2) by sigma. + - Phi is the 2**m-isogeny E1*E2-->Am (m first steps of the chain in dimension 2). + - S1=[2**e]Phi([lamb]P2,[a]sigma(P1)+[b]sigma(P2)). + - S2=[2**e]Phi([mu]P1,[c]sigma(P1)+[d]sigma(P2)). + - T1=[2**(e-m)]Phi([a1]P1-[a2]P2,sigma(P1)). + - T2=[2**(e-m)]Phi([a1]P2+[a2]P1,sigma(P2)). + - (S1,S2,T1,T2) is induced by the image by Phi of a symplectic basis of E1*E2[2**(m+2)] lying + above the symplectic basis of E1*E2[4] outputted by gluing_base_change_matrix_dim2. + + INPUT: + - a1, a2: integers. + - m: integer (number of steps in dimension 2). + - mua2: product mu*a2. + + OUTPUT: + - M: symplectic base change matrix of (*,B_K4) in B. + """ + a1a2_2m=ZZ(a1*a2//2**m) + a22_2m=ZZ(a2**2//2**m) + + Z4=Integers(4) + + C=matrix(Z4,[[-a1a2_2m,a22_2m,a22_2m,a1a2_2m], + [-a22_2m,-a1a2_2m,-a1a2_2m,a22_2m], + [-a22_2m,-a1a2_2m,-a1a2_2m,a22_2m], + [a1a2_2m,-a22_2m,-a22_2m,-a1a2_2m]]) + + D=matrix(Z4,[[1,0,0,0], + [mua2,1,0,-mua2], + [0,0,1,0], + [0,mua2,mua2,1]]) + + M=complete_symplectic_matrix_dim4(C,D,4) + + assert is_symplectic_matrix_dim4(M) + + return M + +def splitting_base_change_matrix_dim4(a1,a2,q,m,M0,A_B,mu=None): + r""" + Let F be the endomorphism of E1^2*E2^2 given by Kani's lemma. Write: + E1^2*E2^2 -- Phi x Phi --> Am^2 -- G --> E1^2*E2^2, + where Phi: E1 x E1 --> Am is a 2**m-isogeny in dimension 2. + Let (U_1,...,U_4,V_1,...,V_4) be a symplectic basis of Am^2[2**(e-m+2)] + such that V_i=Phi x Phi(W_i), where W_1,...,W_4 have order 2**(e+2), lie over ker(F) + and generate an isotropic subgroup: + W_1=([a1]P1-[2^e/a1]P1,[a2]P1,R2,0) + W_2=([a1]Q1,[a2]Q1,S2,0) + W_3=(-[a2]P1,[a1]P1,0,R2) + W_4=(-[a2]Q1,[a1]Q1,0,S2), + with (P1,Q1), a basis of E1[2**(e+2)] and (R2,S2) its image via + sigma: E1 --> E2. Then B:=([2^(e-m)]G(U_1),...,[2^(e-m)]G(U_4),G(V_1),...,G(V_4)) + is a symplectic basis of E1^2*E2^2[4]. + + We assume that ([2^(e-m)]U_1,...,[2^(e-m)]U_4) is the symplectic complement of + ([2^(e-m)]V_1,...,[2^(e-m)]V_4) that has been outputted by + gluing_base_change_matrix_dim2_dim4 for the gluing isogeny on Am^2 + (first 2-isogeny of G). This function computes the base change matrix of B + in the symplectic basis of E1^2*E2^2[4]: + B0=[(T1,0,0,0),(0,T1,0,0),(0,0,T2,0),(0,0,0,T2),(U1,0,0,0),(0,U1,0,0), + (0,0,U2,0),(0,0,0,U2)] + associated to the product Theta structure on E1^2*E2^2. + + INPUT: + - a1,a2,q: integers defining F (q=deg(sigma)). + - m: 2-adic valuation of a2. + - M0: base change matrix of the symplectic basis 2**e*B1 of E1*E2[4] + given by: + B1:=[[(P1,0),(0,R2)],[(Q1,0),(0,lamb*S2)]] + in the canonical symplectic basis: + B0:=[[(T1,0),(0,T2)],[(U1,0),(0,U2)]], + where lamb is the modular inverse of q mod 2**(e+2), so that: + e_{2**(e+2)}(P1,P2)=e_{2**(e+2)}(R1,lamb*R2). + - A_B: 4 first columns (left part) of the symplectic matrix outputted by + gluing_base_change_matrix_dim2_dim4. + - mu, a, b, c, d: integers defining the product Theta structure of Am^2 + given by the four torsion basis [2**(e-m)]*B1 of Am, where: + B1=[[2**m]Phi([2**(m+1)]P2,[a]sigma(P1)+[b]sigma(P2)), + [2**m]Phi([mu]P1,[2**(m+1)]sigma(P1)+[d]sigma(P2)), + Phi([a1]P1-[a2]P2,sigma(P1)), + Phi([a1]P2+[a2]P1,sigma(P2))]. + Only mu is given. + + OUTPUT: The desired base change matrix. + """ + Z4=Integers(4) + + a2_2m=ZZ(a2//2**m) + a12_q_2m=ZZ((a1**2+q)//2**m) + + inv_q=inverse_mod(q,4) + inv_a1=inverse_mod(a1,4) + + lamb=ZZ(2**(m+1)) + if mu==None: + mu=ZZ((1-2**(m+1)*q)*inv_a1) + a=ZZ(2**(m+1)*a2*inv_q) + b=ZZ(-(1+2**(m+1)*a1)*inv_q) + c=ZZ(2**(m+1)) + d=ZZ(-mu*a2*inv_q) + + # Matrix of the four torsion basis of E1^2*E2^2[4] given by + # ([2^(e-m)]G(B1[0],0),[2^(e-m)]G(B1[1],0),[2^(e-m)]G(0,B1[0]),[2^(e-m)]G(0,B1[1]), + # G(B1[2],0),G(B1[3],0),G(0,B1[2]),G(0,B1[3])) in the basis induced by + # [2**e](P1,Q1,R2,[1/q]S2) + M1=matrix(Z4,[[a*q,mu*a1+c*q,0,mu*a2,a12_q_2m,a1*a2_2m,a1*a2_2m,a2*a2_2m], + [0,-mu*a2,a*q,mu*a1+c*q,-a1*a2_2m,-a2*a2_2m,a12_q_2m,a1*a2_2m], + [a1*a,a1*c-mu,-a*a2,-c*a2,0,-a2_2m,-a2_2m,0], + [a2*a,a2*c,a*a1,c*a1-mu,a2_2m,0,0,-a2_2m], + [lamb*a1+b*q,d*q,lamb*a2,0,-a1*a2_2m,a12_q_2m,-a2*a2_2m,a1*a2_2m], + [-lamb*a2,0,lamb*a1+b*q,d*q,a2*a2_2m,-a1*a2_2m,-a1*a2_2m,a12_q_2m], + [(a1*b-lamb)*q,a1*d*q,-b*a2*q,-a2*d*q,a2_2m*q,0,0,-a2_2m*q], + [a2*b*q,a2*d*q,(b*a1-lamb)*q,a1*d*q,0,a2_2m*q,a2_2m*q,0]]) + #A,B,C,D=bloc_decomposition(M1) + #if B.transpose()*A!=A.transpose()*B: + #print("B^T*A!=A^T*B") + #if C.transpose()*D!=D.transpose()*C: + #print("C^T*D!=D^T*C") + #if A.transpose()*D-B.transpose()*C!=identity_matrix(4): + #print(A.transpose()*D-B.transpose()*C) + #print("A^T*D-B^T*C!=I") + #print(M1) + #print(M1.det()) + + # Matrix of ([2^e]G(U_1),...,[2^e]G(U_4)) in the basis induced by + # [2**e](P1,Q1,R2,[1/q]S2) + M_left=M1*A_B + #print(A_B) + #print(M_left) + + # Matrix of (G(V_1),...,G(V_4)) in the basis induced by [2**e](P1,Q1,R2,[1/q]S2) + M_right=matrix(Z4,[[0,0,0,0], + [a2*inv_a1,0,1,0], + [inv_a1,0,0,0], + [0,0,0,0], + [0,1,0,-a2*inv_a1], + [0,0,0,0], + [0,0,0,0], + [0,0,0,q*inv_a1]]) + + # Matrix of the basis induced by [2**e](P1,Q1,R2,[1/q]S2) in the basis + # B0 (induced by T1, U1, T2, U2) + MM0=matrix(Z4,[[M0[0,0],0,M0[0,1],0,M0[0,2],0,M0[0,3],0], + [0,M0[0,0],0,M0[0,1],0,M0[0,2],0,M0[0,3]], + [M0[1,0],0,M0[1,1],0,M0[1,2],0,M0[1,3],0], + [0,M0[1,0],0,M0[1,1],0,M0[1,2],0,M0[1,3]], + [M0[2,0],0,M0[2,1],0,M0[2,2],0,M0[2,3],0], + [0,M0[2,0],0,M0[2,1],0,M0[2,2],0,M0[2,3]], + [M0[3,0],0,M0[3,1],0,M0[3,2],0,M0[3,3],0], + [0,M0[3,0],0,M0[3,1],0,M0[3,2],0,M0[3,3]]]) + + M=MM0*block_matrix(1,2,[M_left,M_right]) + + #A,B,C,D=bloc_decomposition(M) + + #M=complete_symplectic_matrix_dim4(C,D) + + #print(M.det()) + #print(M) + + A,B,C,D=bloc_decomposition(M) + if B.transpose()*A!=A.transpose()*B: + print("B^T*A!=A^T*B") + if C.transpose()*D!=D.transpose()*C: + print("C^T*D!=D^T*C") + if A.transpose()*D-B.transpose()*C!=identity_matrix(4): + print("A^T*D-B^T*C!=I") + assert is_symplectic_matrix_dim4(M) + + return M + +# ============================================================================ # +# Functions for the class KaniEndoHalf (isogeny chain decomposed in two) # +# ============================================================================ # + +def complete_kernel_matrix_F1(a1,a2,q,f): + r"""Computes the symplectic base change matrix of a symplectic basis of the form (*,B_Kp1) + in the symplectic basis of E1^2*E2^2[2**f] given by: + B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)], + [(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]] + where: + - B_Kp1 is a basis of an isotropic subgroup of E1^2*E2^2[2**f] lying above ker(F1). + By convention B_Kp1=[(\tilde{\alpha}_1(P1,0),\Sigma(P1,0)), + (\tilde{\alpha}_1(P2,0),\Sigma(P2,0)), + (\tilde{\alpha}_1(0,P1),\Sigma(0,P1)), + (\tilde{\alpha}_1(0,P2),\Sigma(0,P2))] + - lamb is the inverse of q modulo 2**f. + - (P1,P2) is the canonical basis of E1[2**f]. + - (R1,R2) is the image of (P1,P2) by sigma. + + Input: + - a1, a2, q: Integers such that q+a1**2+a2**2=2**e. + - f: integer determining the accessible 2-torsion in E1 (E1[2**f]). + + Output: + - M: symplectic base change matrix of (*,B_Kp1) in B1. + """ + N=2**f + ZN=Integers(N) + + C=matrix(ZN,[[a1,0,-a2,0], + [a2,0,a1,0], + [1,0,0,0], + [0,0,1,0]]) + + D=matrix(ZN,[[0,a1,0,-a2], + [0,a2,0,a1], + [0,q,0,0], + [0,0,0,q]]) + + assert C.transpose()*D==D.transpose()*C + + M=complete_symplectic_matrix_dim4(C,D,N) + + assert is_symplectic_matrix_dim4(M) + + return M + +def complete_kernel_matrix_F2_dual(a1,a2,q,f): + r"""Computes the symplectic base change matrix of a symplectic basis of the form (*,B_Kp2) + in the symplectic basis of E1^2*E2^2[2**f] given by: + B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)], + [(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]] + where: + - B_Kp2 is a basis of an isotropic subgroup of E1^2*E2^2[2**f] lying above ker(\tilde{F2}). + By convention B_Kp2=[(\alpha_1(P1,0),-\Sigma(P1,0)), + (\alpha_1(P2,0),-\Sigma(P2,0)), + (\alpha_1(0,P1),-\Sigma(0,P1)), + (\alpha_1(0,P2),-\Sigma(0,P2))]. + - lamb is the inverse of q modulo 2**f. + - (P1,P2) is the canonical basis of E1[2**f]. + - (R1,R2) is the image of (P1,P2) by sigma. + + Input: + - a1, a2, q: Integers such that q+a1**2+a2**2=2**e. + - f: integer determining the accessible 2-torsion in E1 (E1[2**f]). + + Output: + - M: symplectic base change matrix of (*,B_Kp2) in B1. + """ + N=2**f + ZN=Integers(N) + + C=matrix(ZN,[[a1,0,a2,0], + [-a2,0,a1,0], + [-1,0,0,0], + [0,0,-1,0]]) + + D=matrix(ZN,[[0,a1,0,a2], + [0,-a2,0,a1], + [0,-q,0,0], + [0,0,0,-q]]) + + + + M=complete_symplectic_matrix_dim4(C,D,N) + + assert is_symplectic_matrix_dim4(M) + + return M + +def matrix_F_dual(a1,a2,q,f): + r""" Computes the matrix of \tilde{F}(B1) in B1, where: + B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)], + [(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]] + as defined in complete_kernel_matrix_F2_dual. + + Input: + - a1, a2, q: Integers such that q+a1**2+a2**2=2**e. + - f: integer determining the accessible 2-torsion in E1 (E1[2**f]). + + Output: + - M: symplectic base change matrix of \tilde{F}(B1) in B1. + """ + N=2**f + ZN=Integers(N) + + M=matrix(ZN,[[a1,-a2,-q,0,0,0,0,0], + [a2,a1,0,-q,0,0,0,0], + [1,0,a1,a2,0,0,0,0], + [0,1,-a2,a1,0,0,0,0], + [0,0,0,0,a1,-a2,-1,0], + [0,0,0,0,a2,a1,0,-1], + [0,0,0,0,q,0,a1,a2], + [0,0,0,0,0,q,-a2,a1]]) + + return M + +def matrix_F(a1,a2,q,f): + r""" Computes the matrix of F(B1) in B1, where: + B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)], + [(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]] + as defined in complete_kernel_matrix_F1. + + Input: + - a1, a2, q: Integers such that q+a1**2+a2**2=2**e. + - f: integer determining the accessible 2-torsion in E1 (E1[2**f]). + + Output: + - M: symplectic base change matrix of \tilde{F}(B1) in B1. + """ + N=2**f + ZN=Integers(N) + + M=matrix(ZN,[[a1,a2,q,0,0,0,0,0], + [-a2,a1,0,q,0,0,0,0], + [-1,0,a1,-a2,0,0,0,0], + [0,-1,a2,a1,0,0,0,0], + [0,0,0,0,a1,a2,1,0], + [0,0,0,0,-a2,a1,0,1], + [0,0,0,0,-q,0,a1,-a2], + [0,0,0,0,0,-q,a2,a1]]) + + return M + +def starting_two_symplectic_matrices(a1,a2,q,f): + r""" + Computes the matrices of two symplectic basis of E1^2*E2^2[2**f] given + by (*,B_Kp1) and (*,B_Kp2) in the basis + B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)], + [(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]] + as defined in complete_kernel_matrix_F1. + + Input: + - a1, a2, q: Integers such that q+a1**2+a2**2=2**e. + - f: integer determining the accessible 2-torsion in E1 (E1[2**f]). + + Output: + - M1, M2: the symplectic base change matrices of (*,B_Kp1) and (*,B_Kp2) in B1. + """ + M1_0=complete_kernel_matrix_F1(a1,a2,q,f) + MatF=matrix_F(a1,a2,q,f) + + # Matrix of an isotropic subgroup of E1^2*E2^2[2**f] lying above ker(\tilde{F2}). + Block_right2=MatF*M1_0[:,[0,1,2,3]] + + N=ZZ(2**f) + + C=Block_right2[[0,1,2,3],:] + D=Block_right2[[4,5,6,7],:] + + assert C.transpose()*D==D.transpose()*C + + # Matrix of the resulting symplectic basis (*,B_Kp2) + M2=complete_symplectic_matrix_dim4(C,D,N) + + MatF_dual=matrix_F_dual(a1,a2,q,f) + + Block_right1=MatF_dual*M2[:,[0,1,2,3]] + + C=Block_right1[[0,1,2,3],:] + D=Block_right1[[4,5,6,7],:] + + A=M1_0[[0,1,2,3],[0,1,2,3]] + B=M1_0[[4,5,6,7],[0,1,2,3]] + + assert C.transpose()*D==D.transpose()*C + assert B.transpose()*A==A.transpose()*B + + # Matrix of the resulting symplectic basis (*,B_Kp1) + M1=block_matrix(1,2,[M1_0[:,[0,1,2,3]],-Block_right1]) + + assert is_symplectic_matrix_dim4(M1) + + A,B,C,D=bloc_decomposition(M1) + a2_div=a2 + m=0 + while a2_div%2==0: + m+=1 + a2_div=a2_div//2 + for j in range(4): + assert (-D[0,j]*a1-C[0,j]*a2-D[2,j])%2**m==0 + assert (C[0,j]*a1-D[0,j]*a2+C[2,j]*q)%2**m==0 + assert (-D[1,j]*a1-C[1,j]*a2-D[3,j])%2**m==0 + assert (C[1,j]*a1-D[1,j]*a2+C[3,j]*q)%2**m==0 + + return M1, M2 + +def gluing_base_change_matrix_dim2_F1(a1,a2,q): + r"""Computes the symplectic base change matrix of a symplectic basis (*,B_K4) of E1*E2[4] + given by the kernel of the dimension 2 gluing isogeny: + B_K4=2**(f-2)[([a1]P1-[a2]P2,R1),([a1]P2+[a2]P1,R2)] + in the basis $2**(f-2)*B1$ given by: + B1:=[[(P1,0),(0,R1)],[(P2,0),(0,[1/q]*R2)]] + where: + - lamb is the inverse of q modulo 2**f. + - (P1,P2) is the canonical basis of E1[2**f]. + - (R1,R2) is the image of (P1,P2) by sigma. + + Input: + - a1, q: integers. + + Output: + - M: symplectic base change matrix of (*,B_K4) in 2**(f-2)*B1. + """ + return gluing_base_change_matrix_dim2(a1,a2,q) + +def gluing_base_change_matrix_dim2_dim4_F1(a1,a2,q,m,M1): + r"""Computes the symplectic base change matrix of the symplectic basis Bp of Am*Am[4] induced + by the image of the symplectic basis (x_1, ..., x_4, y_1, ..., y_4) of E1^2*E2^2[2**(e1+2)] + adapted to ker(F1)=[4] in the basis associated to the product theta-structure + of level 2 of Am*Am: + + B:=[(S1,0),(S2,0),(0,S1),(0,S2),(T1,0),(T2,0),(0,T1),(0,T2)] + + where: + - (P1,Q1) is the canonical basis of E1[2**f]. + - (R2,S2) is the image of (P1,P2) by sigma. + - Phi is the 2**m-isogeny E1*E2-->Am (m first steps of the chain in dimension 2). + - S1=[2**e]Phi([lamb]Q1,[a]sigma(P1)+[b]sigma(Q1)). + - S2=[2**e]Phi([mu]P1,[c]sigma(P1)+[d]sigma(Q1)). + - T1=[2**(e-m)]Phi([a1]P1-[a2]Q1,sigma(P1)). + - T2=[2**(e-m)]Phi([a1]Q1+[a2]P1,sigma(Q1)). + - (S1,S2,T1,T2) is induced by the image by Phi of a symplectic basis of E1*E2[2**(m+2)] lying + above the symplectic basis of E1*E2[4] outputted by gluing_base_change_matrix_dim2. + + INPUT: + - a1, a2, q: integers. + - m: integer (number of steps in dimension 2 and 2-adic valuation of a2). + - M1: matrix of (x_1, ..., x_4, y_1, ..., y_4) in the symplectic basis of E1^2*E2^2[2**(e1+2)] + given by: + + B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R2,0),(0,0,0,R2)], + [(Q1,0,0,0),(0,Q1,0,0),(0,0,[1/q]*S2,0),(0,0,0,[1/q]*S2)]] + + OUTPUT: + - M: symplectic base change matrix of Bp in B. + """ + + inv_a1=inverse_mod(a1,2**(m+2)) + inv_q=inverse_mod(q,2**(m+2)) + lamb=ZZ(2**(m+1)) + mu=ZZ((1-2**(m+1)*q)*inv_a1) + a=ZZ(2**(m+1)*a2*inv_q) + bq=ZZ((-1-2**(m+1)*a1)) + c=ZZ(2**(m+1)) + dq=-ZZ(mu*a2) + + Z4=Integers(4) + + A,B,C,D=bloc_decomposition(M1) + + Ap=matrix(Z4,[[ZZ(-B[0,j]*a1-A[0,j]*a2-B[2,j]) for j in range(4)], + [ZZ(A[0,j]*a1-B[0,j]*a2+A[2,j]*q) for j in range(4)], + [ZZ(-B[1,j]*a1-A[1,j]*a2-B[3,j]) for j in range(4)], + [ZZ(A[1,j]*a1-B[1,j]*a2+A[3,j]*q) for j in range(4)]]) + + Bp=matrix(Z4,[[ZZ(2**m*(B[2,j]*a-A[0,j]*lamb-A[2,j]*bq)) for j in range(4)], + [ZZ(2**m*(B[2,j]*c+B[0,j]*mu-A[2,j]*dq)) for j in range(4)], + [ZZ(2**m*(B[3,j]*a-A[1,j]*lamb-A[3,j]*bq)) for j in range(4)], + [ZZ(2**m*(B[3,j]*c+B[1,j]*mu-A[3,j]*dq)) for j in range(4)]]) + + Cp=matrix(Z4,[[ZZ(ZZ(-D[0,j]*a1-C[0,j]*a2-D[2,j])//(2**m)) for j in range(4)], + [ZZ(ZZ(C[0,j]*a1-D[0,j]*a2+C[2,j]*q)//2**m) for j in range(4)], + [ZZ(ZZ(-D[1,j]*a1-C[1,j]*a2-D[3,j])//2**m) for j in range(4)], + [ZZ(ZZ(C[1,j]*a1-D[1,j]*a2+C[3,j]*q)//2**m) for j in range(4)]]) + + Dp=matrix(Z4,[[ZZ(D[2,j]*a-C[0,j]*lamb-C[2,j]*bq) for j in range(4)], + [ZZ(D[0,j]*mu+D[2,j]*c-C[2,j]*dq) for j in range(4)], + [ZZ(D[3,j]*a-C[1,j]*lamb-C[3,j]*bq) for j in range(4)], + [ZZ(D[1,j]*mu+D[3,j]*c-C[3,j]*dq) for j in range(4)]]) + + M=block_matrix(2,2,[[Ap,Cp],[Bp,Dp]]) + + assert is_symplectic_matrix_dim4(M) + + return M + +def gluing_base_change_matrix_dim2_F2(a1,a2,q): + r"""Computes the symplectic base change matrix of a symplectic basis (*,B_K4) of E1*E2[4] + given by the kernel of the dimension 2 gluing isogeny: + B_K4=2**(f-2)[([a1]P1-[a2]P2,R1),([a1]P2+[a2]P1,R2)] + in the basis $2**(f-2)*B1$ given by: + B1:=[[(P1,0),(0,R1)],[(P2,0),(0,[1/q]*R2)]] + where: + - lamb is the inverse of q modulo 2**f. + - (P1,P2) is the canonical basis of E1[2**f]. + - (R1,R2) is the image of (P1,P2) by sigma. + + Input: + - a1, q: integers. + + Output: + - M: symplectic base change matrix of (*,B_K4) in 2**(f-2)*B1. + """ + + Z4=Integers(4) + + mu=inverse_mod(a1,4) + + A=matrix(Z4,[[0,mu], + [0,0]]) + B=matrix(Z4,[[0,0], + [1,-ZZ(mu*a2)]]) + + C=matrix(Z4,[[ZZ(a1),-ZZ(a2)], + [-1,0]]) + D=matrix(Z4,[[ZZ(a2),ZZ(a1)], + [0,-ZZ(q)]]) + + M=block_matrix([[A,C],[B,D]]) + + assert is_symplectic_matrix_dim2(M) + + return M + +def gluing_base_change_matrix_dim2_dim4_F2(a1,a2,q,m,M2): + r"""Computes the symplectic base change matrix of the symplectic basis Bp of Am*Am[4] induced + by the image of the symplectic basis (x_1, ..., x_4, y_1, ..., y_4) of E1^2*E2^2[2**(e1+2)] + adapted to ker(F1)=[4] in the basis associated to the product theta-structure + of level 2 of Am*Am: + + B:=[(S1,0),(S2,0),(0,S1),(0,S2),(T1,0),(T2,0),(0,T1),(0,T2)] + + where: + - (P1,Q1) is the canonical basis of E1[2**f]. + - (R2,S2) is the image of (P1,P2) by sigma. + - Phi is the 2**m-isogeny E1*E2-->Am (m first steps of the chain in dimension 2). + - S1=[2**e]Phi([lamb]Q1,[a]sigma(P1)+[b]sigma(Q1)). + - S2=[2**e]Phi([mu]P1,[c]sigma(P1)+[d]sigma(Q1)). + - T1=[2**(e-m)]Phi([a1]P1-[a2]Q1,sigma(P1)). + - T2=[2**(e-m)]Phi([a1]Q1+[a2]P1,sigma(Q1)). + - (S1,S2,T1,T2) is induced by the image by Phi of a symplectic basis of E1*E2[2**(m+2)] lying + above the symplectic basis of E1*E2[4] outputted by gluing_base_change_matrix_dim2. + + INPUT: + - a1, a2, q: integers. + - m: integer (number of steps in dimension 2 and 2-adic valuation of a2). + - M2: matrix of (x_1, ..., x_4, y_1, ..., y_4) in the symplectic basis of E1^2*E2^2[2**(e1+2)] + given by: + + B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R2,0),(0,0,0,R2)], + [(Q1,0,0,0),(0,Q1,0,0),(0,0,[1/q]*S2,0),(0,0,0,[1/q]*S2)]] + + OUTPUT: + - M: symplectic base change matrix of Bp in B. + """ + + inv_a1=inverse_mod(a1,2**(m+2)) + inv_q=inverse_mod(q,2**(m+2)) + lamb=ZZ(2**(m+1)) + mu=ZZ((1+2**(m+1)*q)*inv_a1) + a=ZZ(2**(m+1)*a2*inv_q) + bq=ZZ((1+2**(m+1)*a1)) + c=ZZ(2**(m+1)) + dq=-ZZ(mu*a2) + + Z4=Integers(4) + + A,B,C,D=bloc_decomposition(M2) + + Ap=matrix(Z4,[[ZZ(-B[0,j]*a1+A[0,j]*a2+B[2,j]) for j in range(4)], + [ZZ(A[0,j]*a1+B[0,j]*a2-A[2,j]*q) for j in range(4)], + [ZZ(-B[1,j]*a1+A[1,j]*a2+B[3,j]) for j in range(4)], + [ZZ(A[1,j]*a1+B[1,j]*a2-A[3,j]*q) for j in range(4)]]) + + Bp=matrix(Z4,[[ZZ(2**m*(B[2,j]*a-A[0,j]*lamb-A[2,j]*bq)) for j in range(4)], + [ZZ(2**m*(B[2,j]*c+B[0,j]*mu-A[2,j]*dq)) for j in range(4)], + [ZZ(2**m*(B[3,j]*a-A[1,j]*lamb-A[3,j]*bq)) for j in range(4)], + [ZZ(2**m*(B[3,j]*c+B[1,j]*mu-A[3,j]*dq)) for j in range(4)]]) + + Cp=matrix(Z4,[[ZZ(ZZ(-D[0,j]*a1+C[0,j]*a2+D[2,j])//(2**m)) for j in range(4)], + [ZZ(ZZ(C[0,j]*a1+D[0,j]*a2-C[2,j]*q)//2**m) for j in range(4)], + [ZZ(ZZ(-D[1,j]*a1+C[1,j]*a2+D[3,j])//2**m) for j in range(4)], + [ZZ(ZZ(C[1,j]*a1+D[1,j]*a2-C[3,j]*q)//2**m) for j in range(4)]]) + + Dp=matrix(Z4,[[ZZ(D[2,j]*a-C[0,j]*lamb-C[2,j]*bq) for j in range(4)], + [ZZ(D[0,j]*mu+D[2,j]*c-C[2,j]*dq) for j in range(4)], + [ZZ(D[3,j]*a-C[1,j]*lamb-C[3,j]*bq) for j in range(4)], + [ZZ(D[1,j]*mu+D[3,j]*c-C[3,j]*dq) for j in range(4)]]) + + M=block_matrix(2,2,[[Ap,Cp],[Bp,Dp]]) + + assert is_symplectic_matrix_dim4(M) + + return M + +def point_matrix_product(M,L_P,J=None,modulus=None): + r""" + Input: + - M: matrix with (modular) integer values. + - L_P: list of elliptic curve points [P1,P2,R1,R2] such that the rows of M correspond to the vectors + (P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1),(P2,0,0,0),(0,P2,0,0),(0,0,R2,0),(0,0,0,R2). + - J: list of column indices (default, all the columns). + - modulus: order of points in L_P (default, None). + + Output: + - L_ret: list of points corresponding to the columns of M with indices in J. + """ + if modulus==None: + M1=M + else: + Zmod=Integers(modulus) + M1=matrix(Zmod,M) + + if J==None: + J=range(M1.ncols()) + + L_ret=[] + for j in J: + L_ret.append(TuplePoint(M1[0,j]*L_P[0]+M1[4,j]*L_P[1],M1[1,j]*L_P[0]+M1[5,j]*L_P[1], + M1[2,j]*L_P[2]+M1[6,j]*L_P[3],M1[3,j]*L_P[2]+M1[7,j]*L_P[3])) + + return L_ret + + +def kernel_basis(M,ei,mP1,mP2,mR1,mlambR2): + r""" + Input: + - M: matrix of a symplectic basis in the basis + B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)], + [(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]] + as defined in complete_kernel_matrix_F1. + - ei: length of F1 or F2. + - mP1,mP2: canonical basis (P1,P2) of E1[2**f] multiplied by m:=2**(f-ei-2). + - mR1,mlambR2: (mR1,mlambR2)=(m*sigma(P1),m*sigma(P2)), where lamb is the + inverse of q=deg(sigma) modulo 2**f. + + Output: + - Basis of the second symplectic subgroup basis of E1^2*E2^2[2**(ei+2)] induced by M. + """ + modulus=2**(ei+2) + + return point_matrix_product(M,[mP1,mP2,mR1,mlambR2],[4,5,6,7],modulus) + +def base_change_canonical_dim4(P1,P2,R1,R2,q,f,e1,e2): + lamb=inverse_mod(q,2**f) + + lambR2=lamb*R2 + + P1_doubles=[P1] + P2_doubles=[P2] + R1_doubles=[R1] + lambR2_doubles=[lambR2] + + for i in range(f-2): + P1_doubles.append(2*P1_doubles[-1]) + P2_doubles.append(2*P2_doubles[-1]) + R1_doubles.append(2*R1_doubles[-1]) + lambR2_doubles.append(2*lambR2_doubles[-1]) + + # Constructing canonical basis of E1[4] and E2[4]. + _,_,T1,T2,MT=make_canonical(P1_doubles[-1],P2_doubles[-1],4,preserve_pairing=True) + _,_,U1,U2,MU=make_canonical(R1_doubles[-1],lambR2_doubles[-1],4,preserve_pairing=True) + + # Base change matrix of the symplectic basis 2**(f-2)*B1 of E1^2*E2^2[4] in the basis: + # B0:=[[(T1,0,0,0),(0,T1,0,0),(0,0,U1,0),(0,0,0,U1)], + #[(T2,0,0,0),(0,T2,0,0),(0,0,U2,0),(0,0,0,U2)]] + # where B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)], + #[(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]] + Z4=Integers(4) + M0=matrix(Z4,[[MT[0,0],0,0,0,MT[1,0],0,0,0], + [0,MT[0,0],0,0,0,MT[1,0],0,0], + [0,0,MU[0,0],0,0,0,MU[1,0],0], + [0,0,0,MU[0,0],0,0,0,MU[1,0]], + [MT[0,1],0,0,0,MT[1,1],0,0,0], + [0,MT[0,1],0,0,0,MT[1,1],0,0], + [0,0,MU[0,1],0,0,0,MU[1,1],0], + [0,0,0,MU[0,1],0,0,0,MU[1,1]]]) + + return P1_doubles,P2_doubles,R1_doubles,lambR2_doubles,T1,T2,U1,U2,MT,MU,M0 -- cgit v1.2.3-70-g09d2