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from sage.all import *
import itertools
## Index management
@cached_function
def index_to_multindex(k):
r"""
Input:
- k: integer between 0 and 15.
Output: binary decomposition of k.
"""
L_ind=[]
l=k
for i in range(4):
L_ind.append(l%2)
l=l//2
return tuple(L_ind)
@cached_function
def multindex_to_index(*args):
r"""
Input: 4 elements i0,i1,i2,i3 in {0,1}.
Output: k=i0+2*i1+4*i2+8*i3.
"""
if len(args)==4:
i0,i1,i2,i3=args
else:
i0,i1,i2,i3=args[0]
return i0+2*i1+4*i2+8*i3
@cached_function
def scal_prod(i,j):
r"""
Input: Two integers i and j in {0,...,15}.
Output: Scalar product of the bits of i and j mod 2.
"""
return (int(i)&int(j)).bit_count()%2
def act_point(P,I,J):
r"""
Input:
- P: a point with 16 coordinates.
- 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
"""
Q=[]
i=multindex_to_index(I)
j=multindex_to_index(J)
for k in range(16):
ipk=i^k
Q.append((-1)**scal_prod(ipk,j)*P[ipk])
return Q
## Product of theta points
def product_theta_point_dim4(P0,P1,P2,P3):
# Computes the product theta coordinates of a product of 4 elliptic curves.
P=[0 for k in range(16)]
for i0, i1, i2, i3 in itertools.product([0,1],repeat=4):
P[multindex_to_index(i0,i1,i2,i3)]=P0[i0]*P1[i1]*P2[i2]*P3[i3]
return P
def product_theta_point_dim2_dim4(P0,P1):
# Computes the product theta coordinates of a product of 2 abelian surfaces.
P=[0 for k in range(16)]
for i0, i1, i2, i3 in itertools.product([0,1],repeat=4):
P[multindex_to_index(i0,i1,i2,i3)]=P0[i0+2*i1]*P1[i2+2*i3]
return P
## 4-dimensional product Theta point to 1-dimensional Theta points
def product_to_theta_points_dim4(P):
Fp2=P[0].parent()
d_Pi={0:None,1:None,2:None,3:None}
d_index_ratios={0:None,1:None,2:None,3:None}# Index of numertors/denominators to compute the theta points.
for k in range(4):
is_zero=True# theta_1(Pk)=0
for I in itertools.product([0,1],repeat=3):
J=list(I)
J.insert(k,1)
j=multindex_to_index(*J)
if P[j]!=0:
is_zero=False
d_index_ratios[k]=[j^(2**k),j]
break
if is_zero:
d_Pi[k]=(Fp2(1),Fp2(0))
L_num=[]
L_denom=[]
d_index_num_denom={}
for k in range(4):
if d_Pi[k]==None:# Point has non-zero coordinate theta_1
d_index_num_denom[k]=len(L_num)
L_num.append(P[d_index_ratios[k][0]])
L_denom.append(P[d_index_ratios[k][1]])
L_denom=batch_inversion(L_denom)
for k in range(4):
if d_Pi[k]==None:
d_Pi[k]=(L_num[d_index_num_denom[k]]*L_denom[d_index_num_denom[k]],Fp2(1))
return d_Pi[0],d_Pi[1],d_Pi[2],d_Pi[3]
## 4-dimensional product Theta point to 2-dimensional Theta points
def product_to_theta_points_dim4_dim2(P):
Fp2=P[0].parent()
k0=0# Index of the denominator k0=multindex_to_index(I0,J0) and
# we divide by theta_{I0,J0}=theta_{I0}*theta_{J0}!=0
while k0<=15 and P[k0]==0:
k0+=1
i0, j0 = k0%4, k0//4
inv_theta_k0=1/P[k0]
theta_P1=[]
theta_P2=[]
for i in range(4):
if i==i0:
theta_P1.append(1)
else:
theta_P1.append(P[i+4*j0]*inv_theta_k0)
for j in range(4):
if j==j0:
theta_P2.append(1)
else:
theta_P2.append(P[i0+4*j]*inv_theta_k0)
return theta_P1, theta_P2
## Usual theta transformations
@cached_function
def inv_16(F):
return 1/F(16)
def hadamard2(x,y):
return (x+y, x-y)
def hadamard4(x,y,z,t):
x,y=hadamard2(x,y)
z,t=hadamard2(z,t)
return (x+z, y+t, x-z, y-t)
def hadamard8(a,b,c,d,e,f,g,h):
a,b,c,d=hadamard4(a,b,c,d)
e,f,g,h=hadamard4(e,f,g,h)
return (a+e, b+f, c+g, d+h, a-e, b-f, c-g, d-h)
def hadamard16(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p):
a,b,c,d,e,f,g,h=hadamard8(a,b,c,d,e,f,g,h)
i,j,k,l,m,n,o,p=hadamard8(i,j,k,l,m,n,o,p)
return (a+i, b+j, c+k, d+l, e+m, f+n, g+o, h+p, a-i, b-j, c-k, d-l, e-m, f-n, g-o, h-p)
def hadamard_inline(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p):
a,b=a+b,a-b
c,d=c+d,c-d
e,f=e+f,e-f
g,h=g+h,g-h
i,j=i+j,i-j
k,l=k+l,k-l
m,n=m+n,m-n
o,p=o+p,o-p
a,b,c,d=a+c,b+d,a-c,b-d
e,f,g,h=e+g,f+h,e-g,f-h
i,j,k,l=i+k,j+l,i-k,j-l
m,n,o,p=m+o,n+p,m-o,n-p
a,b,c,d,e,f,g,h=a+e, b+f, c+g, d+h, a-e, b-f, c-g, d-h
i,j,k,l,m,n,o,p=i+m,j+n,k+o,l+p,i-m,j-n,k-o,l-p
return (a+i, b+j, c+k, d+l, e+m, f+n, g+o, h+p, a-i, b-j, c-k, d-l, e-m, f-n, g-o, h-p)
def hadamard(P):
return hadamard16(*P)
#return hadamard_inline(*P)
def hadamard_inverse(P):
H_inv_P=[]
C=inv_16(P[0].parent())
for j in range(16):
HP.append(0)
for k in range(16):
if scal_prod(k,j)==0:
H_inv_P[j]+=P[k]
else:
H_inv_P[j]-=P[k]
H_inv_P[j]=H_inv_P[j]*C
return H_inv_P
def squared(P):
return [x**2 for x in P]
def batch_inversion(L):
r"""Does n inversions in 3(n-1)M+1I.
Input:
- L: list of elements to invert.
Output:
- [1/x for x in L]
"""
# Given L=[a0,...,an]
# Computes multiples=[a0, a0.a1, ..., a0...an]
multiples=[L[0]]
for ai in L[1:]:
multiples.append(multiples[-1]*ai)
# Computes inverses=[1/(a0...an),...,1/a0]
inverses=[1/multiples[-1]]
for i in range(1,len(L)):
inverses.append(inverses[-1]*L[-i])
# Finally computes [1/a0,...,1/an]
result=[inverses[-1]]
for i in range(2,len(L)+1):
result.append(inverses[-i]*multiples[i-2])
return result
## Functions to handle zero theta coordinates
def find_zeros(P):
L_ind_zeros=[]
for i in range(16):
if P[i]==0:
L_ind_zeros.append(i)
return L_ind_zeros
def find_translates(L_ind_zeros):
L_ind_non_zero=[]
L_ind_origin=L_ind_zeros.copy()
for i in range(16):
if i not in L_ind_zeros:
L_ind_non_zero.append(i)
L_trans=[]
while L_ind_origin!=[]:
n_target_max=0
ind_trans_max=0
for k in range(16):
trans=[x^k for x in L_ind_origin]
n_target=0
for y in trans:
if y in L_ind_non_zero:
n_target+=1
if n_target>n_target_max:
n_target_max=n_target
ind_trans_max=k
L_trans.append(ind_trans_max)
L_ind_remove=[]
for x in L_ind_origin:
if x^ind_trans_max in L_ind_non_zero:
L_ind_remove.append(x)
for x in L_ind_remove:
L_ind_origin.remove(x)
return L_trans
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