diff options
Diffstat (limited to 'theta_lib/basis_change')
| -rw-r--r-- | theta_lib/basis_change/base_change_dim2.py | 150 | ||||
| -rw-r--r-- | theta_lib/basis_change/base_change_dim4.py | 152 | ||||
| -rw-r--r-- | theta_lib/basis_change/canonical_basis_dim1.py | 76 | ||||
| -rw-r--r-- | theta_lib/basis_change/kani_base_change.py | 975 |
4 files changed, 1353 insertions, 0 deletions
diff --git a/theta_lib/basis_change/base_change_dim2.py b/theta_lib/basis_change/base_change_dim2.py new file mode 100644 index 0000000..4994529 --- /dev/null +++ b/theta_lib/basis_change/base_change_dim2.py @@ -0,0 +1,150 @@ +from sage.all import * + +import itertools + +def complete_symplectic_matrix_dim2(C, D, n=4): + Zn = Integers(n) + + # Compute first row + F = D.stack(-C).transpose() + xi = F.solve_right(vector(Zn, [1, 0])) + + # Rearrange TODO: why? + v = vector(Zn, [xi[2], xi[3], -xi[0], -xi[1]]) + + # Compute second row + F2 = F.stack(v) + yi = F2.solve_right(vector(Zn, [0, 1, 0])) + M = Matrix(Zn, + [ + [xi[0], yi[0], C[0, 0], C[0, 1]], + [xi[1], yi[1], C[1, 0], C[1, 1]], + [xi[2], yi[2], D[0, 0], D[0, 1]], + [xi[3], yi[3], D[1, 0], D[1, 1]], + ], + ) + + return M + +def block_decomposition(M): + """ + Given a 4x4 matrix, return the four 2x2 matrices as blocks + """ + A = M.matrix_from_rows_and_columns([0, 1], [0, 1]) + B = M.matrix_from_rows_and_columns([2, 3], [0, 1]) + C = M.matrix_from_rows_and_columns([0, 1], [2, 3]) + D = M.matrix_from_rows_and_columns([2, 3], [2, 3]) + return A, B, C, D + +def is_symplectic_matrix_dim2(M): + A, B, C, D = block_decomposition(M) + if B.transpose() * A != A.transpose() * B: + return False + if C.transpose() * D != D.transpose() * C: + return False + if A.transpose() * D - B.transpose() * C != identity_matrix(2): + return False + return True + +def base_change_theta_dim2(M, zeta): + r""" + Computes the matrix N (in row convention) of the new level 2 + theta coordinates after a symplectic base change of A[4] given + by M\in Sp_4(Z/4Z). We have: + (theta'_i)=N*(theta_i) + where the theta'_i are the new theta coordinates and theta_i, + the theta coordinates induced by the product theta structure. + N depends on the fourth root of unity zeta=e_4(Si,Ti) (where + (S0,S1,T0,T1) is a symplectic basis if A[4]). + + Inputs: + - M: symplectic base change matrix. + - zeta: a primitive 4-th root of unity induced by the Weil-pairings + of the symplectic basis of A[4]. + + Output: Matrix N of base change of theta-coordinates. + """ + # Split 4x4 matrix into 2x2 blocks + A, B, C, D = block_decomposition(M) + + # Initialise N to 4x4 zero matrix + N = [[0 for _ in range(4)] for _ in range(4)] + + def choose_non_vanishing_index(C, D, zeta): + """ + Choice of reference non-vanishing index (ir0,ir1) + """ + for ir0, ir1 in itertools.product([0, 1], repeat=2): + L = [0, 0, 0, 0] + for j0, j1 in itertools.product([0, 1], repeat=2): + k0 = C[0, 0] * j0 + C[0, 1] * j1 + k1 = C[1, 0] * j0 + C[1, 1] * j1 + + l0 = D[0, 0] * j0 + D[0, 1] * j1 + l1 = D[1, 0] * j0 + D[1, 1] * j1 + + e = -(k0 + 2 * ir0) * l0 - (k1 + 2 * ir1) * l1 + L[ZZ(k0 + ir0) % 2 + 2 * (ZZ(k1 + ir1) % 2)] += zeta ** (ZZ(e)) + + # Search if any L value in L is not zero + if any([x != 0 for x in L]): + return ir0, ir1 + + ir0, ir1 = choose_non_vanishing_index(C, D, zeta) + + for i0, i1, j0, j1 in itertools.product([0, 1], repeat=4): + k0 = A[0, 0] * i0 + A[0, 1] * i1 + C[0, 0] * j0 + C[0, 1] * j1 + k1 = A[1, 0] * i0 + A[1, 1] * i1 + C[1, 0] * j0 + C[1, 1] * j1 + + l0 = B[0, 0] * i0 + B[0, 1] * i1 + D[0, 0] * j0 + D[0, 1] * j1 + l1 = B[1, 0] * i0 + B[1, 1] * i1 + D[1, 0] * j0 + D[1, 1] * j1 + + e = i0 * j0 + i1 * j1 - (k0 + 2 * ir0) * l0 - (k1 + 2 * ir1) * l1 + N[i0 + 2 * i1][ZZ(k0 + ir0) % 2 + 2 * (ZZ(k1 + ir1) % 2)] += zeta ** (ZZ(e)) + + return Matrix(N) + +def montgomery_to_theta_matrix_dim2(zero12,N=identity_matrix(4),return_null_point=False): + r""" + Computes the matrix that transforms Montgomery coordinates on E1*E2 into theta coordinates + with respect to a certain theta structure given by a base change matrix N. + + Input: + - zero12: theta-null point for the product theta structure on E1*E2[2]: + zero12=[zero1[0]*zero2[0],zero1[1]*zero2[0],zero1[0]*zero2[1],zero1[1]*zero2[1]], + where the zeroi are the theta-null points of Ei for i=1,2. + - N: base change matrix from the product theta-structure. + - return_null_point: True if the theta-null point obtained after applying N to zero12 is returned. + + Output: + - Matrix for the change of coordinates: + (X1*X2,Z1*X2,X1*Z2,Z1*Z2)-->(theta_00,theta_10,theta_01,theta_11). + - If return_null_point, the theta-null point obtained after applying N to zero12 is returned. + """ + + Fp2=zero12[0].parent() + + M=zero_matrix(Fp2,4,4) + + for i in range(4): + for j in range(4): + M[i,j]=N[i,j]*zero12[j] + + M2=[] + for i in range(4): + M2.append([M[i,0]+M[i,1]+M[i,2]+M[i,3],-M[i,0]-M[i,1]+M[i,2]+M[i,3],-M[i,0]+M[i,1]-M[i,2]+M[i,3],M[i,0]-M[i,1]-M[i,2]+M[i,3]]) + + if return_null_point: + null_point=[M[i,0]+M[i,1]+M[i,2]+M[i,3] for i in range(4)] + return Matrix(M2), null_point + else: + return Matrix(M2) + +def apply_base_change_theta_dim2(N,P): + Q=[] + for i in range(4): + Q.append(0) + for j in range(4): + Q[i]+=N[i,j]*P[j] + return Q + diff --git a/theta_lib/basis_change/base_change_dim4.py b/theta_lib/basis_change/base_change_dim4.py new file mode 100644 index 0000000..aaf685f --- /dev/null +++ b/theta_lib/basis_change/base_change_dim4.py @@ -0,0 +1,152 @@ +from sage.all import * +import itertools + +from ..theta_structures.theta_helpers_dim4 import multindex_to_index + +def bloc_decomposition(M): + I1=[0,1,2,3] + I2=[4,5,6,7] + + A=M[I1,I1] + B=M[I2,I1] + C=M[I1,I2] + D=M[I2,I2] + + return A,B,C,D + +def mat_prod_vect(A,I): + J=[] + for i in range(4): + J.append(0) + for k in range(4): + J[i]+=A[i,k]*I[k] + return tuple(J) + +def add_tuple(I,J): + K=[] + for k in range(4): + K.append(I[k]+J[k]) + return tuple(K) + +def scal_prod_tuple(I,J): + s=0 + for k in range(4): + s+=I[k]*J[k] + return s + +def red_mod_2(I): + J=[] + for x in I: + J.append(ZZ(x)%2) + return tuple(J) + +def choose_non_vanishing_index(C,D,zeta): + for I0 in itertools.product([0,1],repeat=4): + L=[0 for k in range(16)] + for J in itertools.product([0,1],repeat=4): + CJ=mat_prod_vect(C,J) + DJ=mat_prod_vect(D,J) + e=-scal_prod_tuple(CJ,DJ)-2*scal_prod_tuple(I0,DJ) + + I0pDJ=add_tuple(I0,CJ) + + L[multindex_to_index(red_mod_2(I0pDJ))]+=zeta**(ZZ(e)) + for k in range(16): + if L[k]!=0: + return I0,L + + +def base_change_theta_dim4(M,zeta): + + Z4=Integers(4) + A,B,C,D=bloc_decomposition(M.change_ring(Z4)) + + I0,L0=choose_non_vanishing_index(C,D,zeta) + + + N=[L0]+[[0 for j in range(16)] for i in range(15)] + for I in itertools.product([0,1],repeat=4): + if I!=(0,0,0,0): + AI=mat_prod_vect(A,I) + BI=mat_prod_vect(B,I) + for J in itertools.product([0,1],repeat=4): + CJ=mat_prod_vect(C,J) + DJ=mat_prod_vect(D,J) + + AIpCJ=add_tuple(AI,CJ) + BIpDJ=add_tuple(BI,DJ) + + e=scal_prod_tuple(I,J)-scal_prod_tuple(AIpCJ,BIpDJ)-2*scal_prod_tuple(I0,BIpDJ) + N[multindex_to_index(I)][multindex_to_index(red_mod_2(add_tuple(AIpCJ,I0)))]+=zeta**(ZZ(e)) + + Fp2=zeta.parent() + return matrix(Fp2,N) + +def apply_base_change_theta_dim4(N,P): + Q=[] + for i in range(16): + Q.append(0) + for j in range(16): + Q[i]+=N[i,j]*P[j] + return Q + +def complete_symplectic_matrix_dim4(C,D,n=4): + Zn=Integers(n) + + Col_I4=[matrix(Zn,[[1],[0],[0],[0]]),matrix(Zn,[[0],[1],[0],[0],[0]]), + matrix(Zn,[[0],[0],[1],[0],[0],[0]]),matrix(Zn,[[0],[0],[0],[1],[0],[0],[0]])] + + L_DC=block_matrix([[D.transpose(),-C.transpose()]]) + Col_AB_i=L_DC.solve_right(Col_I4[0]) + A_t=Col_AB_i[[0,1,2,3],0].transpose() + B_t=Col_AB_i[[4,5,6,7],0].transpose() + + for i in range(1,4): + F=block_matrix(2,1,[L_DC,block_matrix(1,2,[B_t,-A_t])]) + Col_AB_i=F.solve_right(Col_I4[i]) + A_t=block_matrix(2,1,[A_t,Col_AB_i[[0,1,2,3],0].transpose()]) + B_t=block_matrix(2,1,[B_t,Col_AB_i[[4,5,6,7],0].transpose()]) + + A=A_t.transpose() + B=B_t.transpose() + + M=block_matrix([[A,C],[B,D]]) + + return M + +def random_symmetric_matrix(n,r): + Zn=Integers(n) + M=zero_matrix(Zn,r,r) + for i in range(r): + M[i,i]=randint(0,n-1) + for i in range(r): + for j in range(i+1,r): + M[i,j]=randint(0,n-1) + M[j,i]=M[i,j] + return M + + +def random_symplectic_matrix(n): + Zn=Integers(n) + + C=random_symmetric_matrix(n,4) + Cp=random_symmetric_matrix(n,4) + + A=random_matrix(Zn,4,4) + while not A.is_invertible(): + A=random_matrix(Zn,4,4) + + tA_inv=A.inverse().transpose() + ACp=A*Cp + return block_matrix([[ACp,A],[C*ACp-tA_inv,C*A]]) + +def is_symplectic_matrix_dim4(M): + A,B,C,D=bloc_decomposition(M) + if B.transpose()*A!=A.transpose()*B: + return False + if C.transpose()*D!=D.transpose()*C: + return False + if A.transpose()*D-B.transpose()*C!=identity_matrix(4): + return False + return True + 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 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]<y_1, ..., y_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]<y_1, ..., y_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 |