def generate_secrets(a0, b0):
# select 3 pairs from 5, they are all independent
# select 4 pairs from 5, they are all not independent, this is certainly
n = len(a0.D)
choose_3_from_5 = list(itertools.combinations(range(5), 3))
count = 0
while True:
print(count)
count += 1
res = []
a1, b1 = generate_vec(n), generate_vec(n)
a2, b2 = generate_vec(n), generate_vec(n)
a3, b3 = generate_vec(n), generate_vec(n)
a4, b4 = generate_vec(n), generate_vec(n)
res.append([a0, a1, a2, a3, a4])
res.append([b0, b1, b2, b3, b4])
fail_flag = False
for choose in choose_3_from_5:
chosen_vecs = set()
for i in range(len(choose)):
chosen_vecs.add(res[0][choose[i]])
chosen_vecs.add(res[1][choose[i]])
if len(chosen_vecs) != 6:
print("p1");
fail_flag = True
break
if not is_independent(chosen_vecs):
print("p2")
fail_flag = True
break
if not fail_flag:
return res
def generate_secrets(a0, b0):
# select 3 pairs from 5, they are all independent
# select 4 pairs from 5, they are all not independent, this is certainly
n = len(a0.D)
choose_3_from_5 = list(itertools.combinations(range(5), 3))
count = 0
while True:
print(count)
count += 1
res = []
a1, b1 = generate_vec(n), generate_vec(n)
a2, b2 = generate_vec(n), generate_vec(n)
a3, b3 = generate_vec(n), generate_vec(n)
a4, b4 = generate_vec(n), generate_vec(n)
res.append([a0, a1, a2, a3, a4])
res.append([b0, b1, b2, b3, b4])
fail_flag = False
for choose in choose_3_from_5:
chosen_vecs = set()
for i in range(len(choose)):
chosen_vecs.add(res[0][choose[i]])
chosen_vecs.add(res[1][choose[i]])
if len(chosen_vecs) != 6:
print("p1")
fail_flag = True
break
if not is_independent(chosen_vecs):
print("p2")
fail_flag = True
break
if not fail_flag:
return res
def my_is_independent(L):
'''
Input:
- L: a list of Vecs
Output:
- boolean: true if the list is linearly independent
Examples:
>>> D = {0, 1, 2}
>>> L = [Vec(D,{0: 1}), Vec(D,{1: 1}), Vec(D,{2: 1}), Vec(D,{0: 1, 1: 1, 2: 1}), Vec(D,{0: 1, 1: 1}), Vec(D,{1: 1, 2: 1})]
>>> my_is_independent(L)
False
>>> my_is_independent(L[:2])
True
>>> my_is_independent(L[:3])
True
>>> my_is_independent(L[1:4])
True
>>> my_is_independent(L[0:4])
False
>>> my_is_independent(L[2:])
False
>>> my_is_independent(L[2:5])
False
>>> L == [Vec(D,{0: 1}), Vec(D,{1: 1}), Vec(D,{2: 1}), Vec(D,{0: 1, 1: 1, 2: 1}), Vec(D,{0: 1, 1: 1}), Vec(D,{1: 1, 2: 1})]
True
'''
return is_independent(L)
def superset_basis(C, T):
'''
Input:
- C: linearly independent set of Vecs
- T: set of Vecs such that every Vec in S is in Span(T)
Output:
Linearly independent set S consisting of all Vecs in C and some in T
such that the span of S is the span of T (i.e. S is a basis for the span
of T).
Example:
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = superset_basis({a0, a3}, {a0, a1, a2})
>>> a0 in sb and a3 in sb
True
>>> is_independent(sb)
True
>>> all(x in [a0,a1,a2,a3] for x in sb)
True
'''
start, tmp = C.copy(), C.copy()
for x in T:
tmp.add(x)
if is_independent(tmp): start.add(x)
else: tmp.discard(x)
return start
def superset_basis(C, T):
'''
Input:
- C: linearly independent set of Vecs
- T: set of Vecs such that every Vec in S is in Span(T)
Output:
Linearly independent set S consisting of all Vecs in C and some in T
such that the span of S is the span of T (i.e. S is a basis for the span
of T).
Example:
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = superset_basis({a0, a3}, {a0, a1, a2})
>>> a0 in sb and a3 in sb
True
>>> is_independent(sb)
True
>>> all(x in [a0,a1,a2,a3] for x in sb)
True
'''
start, tmp = C.copy(), C.copy()
for x in T:
tmp.add(x)
if is_independent(tmp): start.add(x)
else: tmp.discard(x)
return start
def superset_basis(C, T):
'''
Input:
- C: linearly independent set of Vecs
- T: set of Vecs such that every Vec in C is in Span(T)
Output:
Linearly independent set S consisting of all Vecs in C and some in T
such that the span of S is the span of T (i.e. S is a basis for the span
of T).
Example:
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = superset_basis({a0, a3}, {a0, a1, a2})
>>> a0 in sb and a3 in sb
True
>>> is_independent(sb)
True
>>> all(x in [a0,a1,a2,a3] for x in sb)
True
'''
S = {x for x in C}
for v in T:
S.add(v)
if is_independent(S) == False:
S.remove(v)
return S
def superset_basis(C, T):
'''
Input:
- C: linearly independent set of Vecs
- T: set of Vecs such that every Vec in C is in Span(T)
Output:
Linearly independent set S consisting of all Vecs in C and some in T
such that the span of S is the span of T (i.e. S is a basis for the span
of T).
Example:
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = superset_basis({a0, a3}, {a0, a1, a2})
>>> a0 in sb and a3 in sb
True
>>> is_independent(sb)
True
>>> all(x in [a0,a1,a2,a3] for x in sb)
True
'''
S={x for x in C}
for v in T:
S.add(v)
if is_independent(S)==False:
S.remove(v)
return S
def subset(T):
A = []
for i in T:
if i not in A:
A.append(i)
if not is_independent(A):
A.pop()
return A
def is_invertible(M):
'''
input: A matrix, M
outpit: A boolean indicating if M is invertible.
>>> M = Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): 0, (1, 2): 1, (3, 2): 0, (0, 0): 1, (3, 3): 4, (3, 0): 0, (3, 1): 0, (1, 1): 2, (2, 1): 0, (0, 2): 1, (2, 0): 0, (1, 3): 0, (2, 3): 1, (2, 2): 3, (1, 0): 0, (0, 3): 0})
>>> is_invertible(M)
True
>>> M1 = Mat(({0,1,2},{0,1,2}),{(0,0):1,(0,2):2,(1,2):3,(2,2):4})
>>> is_invertible(M1)
False
'''
return is_independent([v for v in mat2coldict(M).values()]) and (len(M.D[0]) == len(M.D[1]))
def is_invertible(M):
'''
input: A matrix, M
outpit: A boolean indicating if M is invertible.
>>> M = Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): 0, (1, 2): 1, (3, 2): 0, (0, 0): 1, (3, 3): 4, (3, 0): 0, (3, 1): 0, (1, 1): 2, (2, 1): 0, (0, 2): 1, (2, 0): 0, (1, 3): 0, (2, 3): 1, (2, 2): 3, (1, 0): 0, (0, 3): 0})
>>> is_invertible(M)
True
>>> M1 = Mat(({0,1,2},{0,1,2}),{(0,0):1,(0,2):2,(1,2):3,(2,2):4})
>>> is_invertible(M1)
False
'''
cols = mat2coldict(M)
return is_independent(cols.values())
def is_invertible(M):
'''
input: A matrix, M
outpit: A boolean indicating if M is invertible.
>>> M = Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): 0, (1, 2): 1, (3, 2): 0, (0, 0): 1, (3, 3): 4, (3, 0): 0, (3, 1): 0, (1, 1): 2, (2, 1): 0, (0, 2): 1, (2, 0): 0, (1, 3): 0, (2, 3): 1, (2, 2): 3, (1, 0): 0, (0, 3): 0})
>>> is_invertible(M)
True
>>> M1 = Mat(({0,1,2},{0,1,2}),{(0,0):1,(0,2):2,(1,2):3,(2,2):4})
>>> is_invertible(M1)
False
'''
cols = mat2coldict(M)
return is_independent(cols.values())
def subset_basis(T):
'''
Input:
- T: a set of Vecs
Output:
- set S containing Vecs from T that is a basis for Span T.
Examples:
The following tests use the procedure is_independent, provided in module independence
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = subset_basis({a0, a1, a2, a3})
>>> len(sb)
3
>>> all(v in [a0, a1, a2, a3] for v in sb)
True
>>> is_independent(sb)
True
>>> b0 = Vec({0,1,2,3},{0:2,1:2,3:4})
>>> b1 = Vec({0,1,2,3},{0:1,1:1})
>>> b2 = Vec({0,1,2,3},{2:3,3:4})
>>> b3 = Vec({0,1,2,3},{3:3})
>>> sb = subset_basis({b0, b1, b2, b3})
>>> len(sb)
3
>>> all(v in [b0, b1, b2, b3] for v in sb)
True
>>> is_independent(sb)
True
>>> D = {'a','b','c','d'}
>>> c0, c1, c2, c3, c4 = Vec(D,{'d': one, 'c': one}), Vec(D,{'d': one, 'a': one, 'c': one, 'b': one}), Vec(D,{'a': one}), Vec(D,{}), Vec(D,{'d': one, 'a': one, 'b': one})
>>> subset_basis({c0,c1,c2,c3,c4}) == {c0,c1,c2,c4}
True
'''
res = set()
for x in T:
res.add(x)
if is_independent(res) == False:
res.remove(x)
return res
def subset_basis(T):
'''
Input:
- T: a set of Vecs
Output:
- set S containing Vecs from T that is a basis for Span T.
Examples:
The following tests use the procedure is_independent, provided in module independence
>>> from vec import Vec
>>> from independence import is_independent
>>> a0 = Vec({'a','b','c','d'}, {'a':1})
>>> a1 = Vec({'a','b','c','d'}, {'b':1})
>>> a2 = Vec({'a','b','c','d'}, {'c':1})
>>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
>>> sb = subset_basis({a0, a1, a2, a3})
>>> len(sb)
3
>>> all(v in [a0, a1, a2, a3] for v in sb)
True
>>> is_independent(sb)
True
>>> b0 = Vec({0,1,2,3},{0:2,1:2,3:4})
>>> b1 = Vec({0,1,2,3},{0:1,1:1})
>>> b2 = Vec({0,1,2,3},{2:3,3:4})
>>> b3 = Vec({0,1,2,3},{3:3})
>>> sb = subset_basis({b0, b1, b2, b3})
>>> len(sb)
3
>>> all(v in [b0, b1, b2, b3] for v in sb)
True
>>> is_independent(sb)
True
>>> D = {'a','b','c','d'}
>>> c0, c1, c2, c3, c4 = Vec(D,{'d': one, 'c': one}), Vec(D,{'d': one, 'a': one, 'c': one, 'b': one}), Vec(D,{'a': one}), Vec(D,{}), Vec(D,{'d': one, 'a': one, 'b': one})
>>> subset_basis({c0,c1,c2,c3,c4}) == {c0,c1,c2,c4}
True
'''
res = set()
for x in T:
res.add(x)
if is_independent(res)==False:
res.remove(x)
return res
