def lin_comb_mat_vec_mult(M, v):
assert M.D[1] == v.D
import matutil
from matutil import mat2coldict
mat2col = mat2coldict(M)
return sum([getitem(v, key) * mat2col[key] for key in v.D])
def lin_comb_vec_mat_mult(v, M):
assert v.D == M.D[0]
import matutil
from matutil import mat2rowdict
mat2row = mat2rowdict(M)
return sum([getitem(v, key) * mat2row[key] for key in v.D])
def find_a_and_b(v, roots, N):
'''
Input:
- a {0,1,..., n-1}-vector v over GF(2) where n = len(roots)
- a list roots of integers
- an integer N to factor
Output:
a pair (a,b) of integers
such that a*a-b*b is a multiple of N
(if v is correctly chosen)
Example:
>>> roots = [51, 52, 53, 58, 61, 62, 63, 67, 68, 71, 77, 79]
>>> N = 2419
>>> v = Vec({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11},{1: one, 2: one, 11: one, 5: one})
>>> find_a_and_b(v, roots, N)
(13498888, 778050)
>>> v = Vec({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11},{0: 0, 1: 0, 10: one, 2: one})
>>> find_a_and_b(v, roots, N)
(4081, 1170)
'''
alist = [roots[i] for i in v.D if getitem(v, i) == one]
a = prod(alist)
c = prod([x * x - N for x in alist])
b = intsqrt(c)
assert b * b == c
return (a, b)
def find_a_and_b(v, roots, N):
"""
Input:
- a {0,1,..., n-1}-vector v over GF(2) where n = len(roots)
- a list roots of integers
- an integer N to factor
Output:
a pair (a,b) of integers
such that a*a-b*b is a multiple of N
(if v is correctly chosen)
Example:
>>> roots = [51, 52, 53, 58, 61, 62, 63, 67, 68, 71, 77, 79]
>>> N = 2419
>>> v = Vec({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11},{1: one, 2: one, 11: one, 5: one})
>>> find_a_and_b(v, roots, N)
(13498888, 778050)
>>> v = Vec({0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11},{0: 0, 1: 0, 10: one, 2: one})
>>> find_a_and_b(v, roots, N)
(4081, 1170)
"""
alist = [roots[i] for i in v.D if getitem(v, i) == one]
a = prod(alist)
c = prod([x * x - N for x in alist])
b = intsqrt(c)
assert b * b == c
return (a, b)
def vec_select(veclist, k):
'''
>>> D = {'a','b','c'}
>>> v1 = Vec(D, {'a': 1})
>>> v2 = Vec(D, {'a': 0, 'b': 1})
>>> v3 = Vec(D, { 'b': 2})
>>> v4 = Vec(D, {'a': 10, 'b': 10})
>>> vec_select([v1, v2, v3, v4], 'a') == [Vec(D,{'b': 1}), Vec(D,{'b': 2})]
True
'''
return [v for v in veclist if getitem(v, k) == 0]
def vec_select(veclist, k):
'''
>>> D = {'a','b','c'}
>>> v1 = Vec(D, {'a': 1})
>>> v2 = Vec(D, {'a': 0, 'b': 1})
>>> v3 = Vec(D, { 'b': 2})
>>> v4 = Vec(D, {'a': 10, 'b': 10})
>>> vec_select([v1, v2, v3, v4], 'a') == [Vec(D,{'b': 1}), Vec(D,{'b': 2})]
True
'''
return [v for v in veclist if getitem(v,k)==0]
def lin_comb_vec_mat_mult(v, M):
assert (v.D == M.D[0])
w = mat2rowdict(M)
s = 0
l = []
for (i, j) in w.items():
scalar = vec.getitem(v, i)
s += scalar * j
l.append(s)
s = 0
r = sum(l)
return r
def lin_comb_mat_vec_mult(M, v):
assert (M.D[1] == v.D)
w = mat2coldict(M)
s = 0
l = []
for (i, j) in w.items():
scalar = vec.getitem(v, i)
s += scalar * j
l.append(s)
s = 0
r = sum(l)
return r
def vec_select(veclist, k):
'''
>>> D = {'a','b','c'}
>>> v1 = Vec(D, {'a': 1})
>>> v2 = Vec(D, {'a': 0, 'b': 1})
>>> v3 = Vec(D, { 'b': 2})
>>> v4 = Vec(D, {'a': 10, 'b': 10})
>>> vec_select([v1, v2, v3, v4], 'a') == [Vec(D,{'b': 1}), Vec(D,{'b': 2})]
True
'''
from vec import getitem
return [x for x in veclist if getitem(x,k)==0]
def lin_comb_vec_mat_mult(v, M):
assert(v.D == M.D[0])
w = mat2rowdict(M)
s=0
l = []
for (i,j) in w.items():
scalar = vec.getitem(v, i)
s += scalar * j
l.append(s)
s=0
r = sum(l)
return r
def lin_comb_mat_vec_mult(M, v):
assert(M.D[1] == v.D)
w = mat2coldict(M)
s=0
l = []
for (i,j) in w.items():
scalar = vec.getitem(v, i)
s += scalar * j
l.append(s)
s=0
r = sum(l)
return r
def find_a_and_b(v, roots, N):
'''
Input:
- a {0,1,..., n-1}-vector v over GF(2) where n = len(roots)
- a list roots of integers
- an integer N to factor
Output:
a pair (a,b) of integers
such that a*a-b*b is a multiple of N
(if v is correctly chosen)
'''
alist = [roots[s] for s in v.D if getitem(v,s) != 0]
a = prod(alist)
c = prod([k**2 - N for k in alist])
b = intsqrt(c)
assert b**2 == c
return (a,b)
def matrix_vector_mul(M, v):
"Returns the product of matrix M and vector v"
assert M.D[1] == v.D
return Vec(M.D[0], { i : sum(getitem(M, (i, j)) * vec.getitem(v, j) for j in M.D[1]) for i in M.D[0] })
def vector_matrix_mul(v, M):
"Returns the product of vector v and matrix M"
assert M.D[0] == v.D
return Vec(M.D[1], { i : sum(vec.getitem(v, j) * getitem(M, (j, i)) for j in M.D[0]) for i in M.D[1] })
def lin_comb_vec_mat_mult(v, M):
assert(v.D == M.D[0])
return sum(vec.getitem(v,i) * mat2rowdict(M)[i] for i in v.D)
def lin_comb_mat_vec_mult(M, v):
assert(M.D[1] == v.D)
return sum(vec.getitem(v,i) * mat2coldict(M)[i] for i in v.D)