def intersect(i, j, **gb_opts):
"""
This functions intersects two ideals. The first ring variable is used as helper variable for this
intersection. It is assumed, that it doesn't occur in the ideals, and that we have an elimination ordering
for this variables. Both assumptions are checked.
>>> from polybori.frontend import declare_ring
>>> from polybori import Block
>>> r=declare_ring(Block("x", 1000), globals())
>>> x = r.variable
>>> intersect([x(1),x(2)+1],[x(1),x(2)])
[x(1)]
"""
if not i or not j:
return []
uv = used_vars_set(i) * used_vars_set(j)
t = iter(i).next().ring().variable(0)
if uv.reducible_by(t):
raise ValueError, \
"First ring variable has to be reserved as helper variable t"
if not t > uv:
raise ValueError, "need elimination ordering for first ring variable"
gb = groebner_basis(
list(chain((t * p for p in i), ((1 + t) * p for p in j))), **gb_opts)
return [p for p in gb if p.navigation().value() > t.index()]
def intersect(i, j, **gb_opts):
"""
This functions intersects two ideals. The first ring variable is used as helper variable for this
intersection. It is assumed, that it doesn't occur in the ideals, and that we have an elimination ordering
for this variables. Both assumptions are checked.
>>> from polybori.frontend import declare_ring
>>> from polybori import Block
>>> r=declare_ring(Block("x", 1000), globals())
>>> x = r.variable
>>> intersect([x(1),x(2)+1],[x(1),x(2)])
[x(1)]
"""
if not i or not j:
return []
uv = used_vars_set(i) * used_vars_set(j)
t = iter(i).next().ring().variable(0)
if uv.reducible_by(t):
raise ValueError, \
"First ring variable has to be reserved as helper variable t"
if not t > uv:
raise ValueError, "need elimination ordering for first ring variable"
gb = groebner_basis(list(chain((t * p for p in i), ((1 + t) * p for p in j
))), **gb_opts)
return [p for p in gb if p.navigation().value() > t.index()]
def ideals(self):
def sort_key(p):
return p.navigation().value()
def my_sort(l):
return sorted(l, key=sort_key)
#self.intermediate_equations = self.gauss(self.intermediate_equations)
tail_variables = set(used_vars_set([e.mapped_to for e in chain(self.next_state_equations, self.intermediate_equations)]).variables())
to_delete=[]
for v in self.deletion_candidates:
if not v in tail_variables:
to_delete.append(v)
to_delete=set(to_delete)
self.intermediate_equations=[e for e in self.intermediate_equations
if not e.variable in to_delete]
#we don't delete the variable itself, as we will confuse gluemultipliers,that looks on the lengths of self.intermediate...
def equations(determining_equations):
return [e.equation for e in determining_equations]
ideal_state=[]
ideal_next_state = equations(self.next_state_equations)
ideal_intermediate = equations(self.intermediate_equations)
if self.initialize!="noinit":
if self.initialize=="zero":
initializer = zero_fun
else:
initializer = random_bit
for v in variables:
if str(v)[:1]=="s":
ideal_state.append(v+initializer())
return [
my_sort(self.apply_map(i)) for i in
(ideal_state,ideal_intermediate, ideal_next_state)]
def add_polynomial_to_cluster(self, p):
self.cluster.add(p)
self.used_variables_cluster = set(
used_vars_set(self.cluster).variables())
self.number_of_used_variables_in_cluster = len(
self.used_variables_cluster)
self.adjust_variables_introduction_mapping(
self.used_variables_of_polynomial[p])
def eliminate_ll_ranked(ll_system, to_reduce, reduction_function=ll_red_nf_noredsb, reduce_ll_system=False, prot=False):
assert ll_system
from_ring = ll_system[0].ring()
ll_ranks = rank(ll_system)
add_vars = set(used_vars_set(to_reduce).variables()).difference(ll_ranks.keys())
for v in add_vars:
ll_ranks[v] = -1
# pushing variables ignored by ll to the front means,
# that the routines will quickly eliminate them
# and they won't give any overhead
def sort_key(v):
return (ll_ranks[v], v.index())
sorted_vars = sorted(ll_ranks.keys(), key=sort_key)
def var_index(v):
return iter(Monomial(v).variables()).next().index()
# sorted_var_indices=[var_index(v) for v in sorted_vars]
to_ring = Ring(len(sorted_vars))
map_back_indices = dict([(i, var_index(v)) for (i, v) in enumerate(sorted_vars)])
map_from_indices = dict([(var_index(v), i) for (i, v) in enumerate(sorted_vars)])
# dict([(v,k) for (k,v) in enumerate(sorted_var_indices)])
var_names = [str(v) for v in sorted_vars]
try:
for (i, v) in enumerate(sorted_vars):
assert var_names[i] == str(v), (var_names[i], v, var_index(v), i)
# _set_variable_name(to_ring, i, var_names[i] + "TO")
finally:
pass
try:
map_from_vec = construct_map_by_indices(to_ring, map_from_indices)
finally:
pass
map_back_vec = construct_map_by_indices(from_ring, map_back_indices)
def map_from(p):
res = substitute_variables(to_ring, map_from_vec, p)
# assert str(p)==str(res), (str(p), str(res), list(map_from_vec), list(map_back_vec))
return res
def map_back(p):
return substitute_variables(from_ring, map_back_vec, p)
try:
ll_opt_encoded = ll_encode([map_from(p) for p in ll_system], prot=False, reduce=reduce_ll_system)
def llnf(p):
return map_back(reduction_function(map_from(p), ll_opt_encoded))
opt_eliminated = [llnf(p) for p in to_reduce]
finally:
pass
return (llnf, opt_eliminated)
def want_la():
if not I:
return False
n_used_vars = None
bound = None
if iter(I).next().ring().has_degree_order():
new_bound = 200
n_used_vars = used_vars_set(I, bound=new_bound).deg()
if n_used_vars < new_bound:
return True
bound = new_bound
if dense_system(I):
new_bound = 100
if not (bound and new_bound < bound):
n_used_vars = used_vars_set(I, bound=new_bound).deg()
bound = new_bound
if n_used_vars < bound:
return True
return False
def want_la():
if not I:
return False
n_used_vars = None
bound = None
if iter(I).next().ring().has_degree_order():
new_bound = 200
n_used_vars = used_vars_set(I, bound=new_bound).deg()
if n_used_vars < new_bound:
return True
bound = new_bound
if dense_system(I):
new_bound = 100
if not (bound and new_bound < bound):
n_used_vars = used_vars_set(I, bound=new_bound).deg()
bound = new_bound
if n_used_vars < bound:
return True
return False
def ll_is_good(I):
lex_lead = set()
for p in I:
if not p.is_zero():
m = p.lex_lead()
if m.deg() == 1:
lex_lead.add(iter(m.variables()).next().index())
if len(lex_lead) >= 0.8 * len(I):
uv = used_vars_set(I).deg() # don't use len here, which will yield 1
if len(lex_lead) > 0.9 * uv:
if uv - len(lex_lead) > 16:
return "llfirstonthefly"
else:
return "llfirst"
return False
def ll_is_good(I):
lex_lead = set()
for p in I:
if not p.is_zero():
m = p.lex_lead()
if m.deg() == 1:
lex_lead.add(iter(m.variables()).next().index())
if len(lex_lead) >= 0.8 * len(I):
uv = used_vars_set(I).deg() # don't use len here, which will yield 1
if len(lex_lead) > 0.9 * uv:
if uv - len(lex_lead) > 16:
return "llfirstonthefly"
else:
return "llfirst"
return False
def variety_size_from_gb(I):
"""
>>> r=Ring(100)
>>> x = r.variable
>>> variety_size_from_gb([])
1
>>> variety_size_from_gb([Polynomial(0, r)])
1
>>> variety_size_from_gb([Polynomial(1, r)])
0.0
>>> variety_size_from_gb([x(1)])
1.0
>>> variety_size_from_gb([x(1), x(2)])
1.0
>>> variety_size_from_gb([x(1), x(2)*x(3)])
3.0
>>> variety_size_from_gb([x(1), x(1)*x(4), x(2)*x(3)])
6.0
>>> variety_size_from_gb([x(1)*x(2), x(2)*x(3)])
5.0
>>> mons = [Monomial([r.variable(i) for i in xrange(100) if i!=j])\
for j in xrange(100)]
>>> variety_size_from_gb(mons)
1.2676506002282294e+30
"""
I = [Polynomial(p) for p in I]
I = [p for p in I if not p.is_zero()]
if len(I) == 0:
return 1
## # TODO Here's something wrong! See the example with 5 solutions.
## # (reverting for now)
## number_of_used_vars = used_vars_set(I).deg()
## leads = set([p.lead() for p in I])
## minimal_leads = BooleSet(leads).minimal_elements()
## number_of_used_vars_minimal_leads =\
## minimal_leads.vars().deg()
## standard_monomials =\
## minimal_leads.include_divisors().diff(minimal_leads)
## return standard_monomials.size_double()*\
## 2**(number_of_used_vars-number_of_used_vars_minimal_leads)
sm = Monomial(used_vars_set(I)).divisors()
for p in I:
m = p.lead()
sm = sm.diff(sm.multiples_of(m))
return sm.size_double()
def dimacs_cnf(self, polynomial_system):
r"""
>>> from polybori import *
>>> r = declare_ring(["x", "y", "z"], dict())
>>> e = CryptoMiniSatEncoder(r)
>>> e.dimacs_cnf([r.variable(0)*r.variable(1)*r.variable(2)])
'c cnf generated by PolyBoRi\np cnf 3 1\n-1 -2 -3 0\nc g 1 x*y*z\nc v 1 x\nc v 2 y\nc v 3 z'
>>> e.dimacs_cnf([r.variable(1)+r.variable(0)])
'c cnf generated by PolyBoRi\np cnf 3 1\nx1 2 0\nc g 2 x + y\nc v 1 x\nc v 2 y'
>>> e.dimacs_cnf([r.variable(0)*r.variable(1)*r.variable(2), r.variable(1)+r.variable(0)])
'c cnf generated by PolyBoRi\np cnf 3 2\n-1 -2 -3 0\nc g 3 x*y*z\nx1 2 0\nc g 4 x + y\nc v 1 x\nc v 2 y\nc v 3 z'
"""
uv=list(used_vars_set(polynomial_system).variables())
res=super(CryptoMiniSatEncoder, self).dimacs_cnf(polynomial_system)
res=res+"\n"+"\n".join(["c v %s %s"% (self.to_dimacs_index(v), v) for v in uv])
return res
def variety_size_from_gb(I):
"""
>>> r=Ring(100)
>>> x = r.variable
>>> variety_size_from_gb([])
1
>>> variety_size_from_gb([Polynomial(0, r)])
1
>>> variety_size_from_gb([Polynomial(1, r)])
0.0
>>> variety_size_from_gb([x(1)])
1.0
>>> variety_size_from_gb([x(1), x(2)])
1.0
>>> variety_size_from_gb([x(1), x(2)*x(3)])
3.0
>>> variety_size_from_gb([x(1), x(1)*x(4), x(2)*x(3)])
6.0
>>> variety_size_from_gb([x(1)*x(2), x(2)*x(3)])
5.0
>>> mons = [Monomial([r.variable(i) for i in xrange(100) if i!=j])\
for j in xrange(100)]
>>> variety_size_from_gb(mons)
1.2676506002282294e+30
"""
I = [Polynomial(p) for p in I]
I = [p for p in I if not p.is_zero()]
if len(I) == 0:
return 1
## # TODO Here's something wrong! See the example with 5 solutions.
## # (reverting for now)
## number_of_used_vars = used_vars_set(I).deg()
## leads = set([p.lead() for p in I])
## minimal_leads = BooleSet(leads).minimal_elements()
## number_of_used_vars_minimal_leads =\
## minimal_leads.vars().deg()
## standard_monomials =\
## minimal_leads.include_divisors().diff(minimal_leads)
## return standard_monomials.size_double()*\
## 2**(number_of_used_vars-number_of_used_vars_minimal_leads)
sm = Monomial(used_vars_set(I)).divisors()
for p in I:
m = p.lead()
sm = sm.diff(sm.multiples_of(m))
return sm.size_double()
def dimacs_cnf(self, polynomial_system):
r"""
>>> from polybori import *
>>> r = declare_ring(["x", "y", "z"], dict())
>>> e = CryptoMiniSatEncoder(r)
>>> e.dimacs_cnf([r.variable(0)*r.variable(1)*r.variable(2)])
'c cnf generated by PolyBoRi\np cnf 3 1\n-1 -2 -3 0\nc g 1 x*y*z\nc v 1 x\nc v 2 y\nc v 3 z'
>>> e.dimacs_cnf([r.variable(1)+r.variable(0)])
'c cnf generated by PolyBoRi\np cnf 3 1\nx1 2 0\nc g 2 x + y\nc v 1 x\nc v 2 y'
>>> e.dimacs_cnf([r.variable(0)*r.variable(1)*r.variable(2), r.variable(1)+r.variable(0)])
'c cnf generated by PolyBoRi\np cnf 3 2\n-1 -2 -3 0\nc g 3 x*y*z\nx1 2 0\nc g 4 x + y\nc v 1 x\nc v 2 y\nc v 3 z'
"""
uv = list(used_vars_set(polynomial_system).variables())
res = super(CryptoMiniSatEncoder, self).dimacs_cnf(polynomial_system)
res = res + "\n" + "\n".join(["c v %s %s" % (self.to_dimacs_index(v),
v) for v in uv])
return res
def ideals(self):
def sort_key(p):
return p.navigation().value()
def my_sort(l):
return sorted(l, key=sort_key)
#self.intermediate_equations = self.gauss(self.intermediate_equations)
tail_variables = set(
used_vars_set([
e.mapped_to for e in chain(self.next_state_equations,
self.intermediate_equations)
]).variables())
to_delete = []
for v in self.deletion_candidates:
if not v in tail_variables:
to_delete.append(v)
to_delete = set(to_delete)
self.intermediate_equations = [
e for e in self.intermediate_equations
if not e.variable in to_delete
]
# we don't delete the variable itself, as we will confuse
# gluemultipliers,that looks on the lengths of self.intermediate...
def equations(determining_equations):
return [e.equation for e in determining_equations]
ideal_state = []
ideal_next_state = equations(self.next_state_equations)
ideal_intermediate = equations(self.intermediate_equations)
if self.initialize != "noinit":
if self.initialize == "zero":
initializer = zero_fun
else:
initializer = random_bit
for v in variables:
if str(v)[:1] == "s":
ideal_state.append(v + initializer())
return [
my_sort(self.apply_map(i))
for i in (ideal_state, ideal_intermediate, ideal_next_state)
]
def __init__(self, ideal, determination_modifier=1):
if len(ideal) == 0:
raise ValueError, 'ideal generators list should be non empty'
self.ideal = ideal
self.determination_modifier = determination_modifier
self.used_variables_ideal = used_vars_set(ideal)
self.number_of_used_variables_in_ideal = len(self.used_variables_ideal)
self.used_variables_of_polynomial = dict([
(p, set(p.vars_as_monomial().variables())) for p in ideal
])
self.variables_introduction_mapping = dict()
self.cluster = set()
self.used_variables_cluster = set()
self.build_variables_usage()
self.initialize_variables_introduction_mapping()
def __init__(self, ideal, determination_modifier=1):
if len(ideal) == 0:
raise ValueError, 'ideal generators list should be non empty'
self.ideal = ideal
self.determination_modifier = determination_modifier
self.used_variables_ideal = used_vars_set(ideal)
self.number_of_used_variables_in_ideal = len(self.used_variables_ideal)
self.used_variables_of_polynomial = dict(
[(p, set(p.vars_as_monomial().variables())) for p in ideal])
self.variables_introduction_mapping = dict()
self.cluster = set()
self.used_variables_cluster = set()
self.build_variables_usage()
self.initialize_variables_introduction_mapping()
def branch(strat,trace):
print "branching"
index=strat.suggestPluginVariable();
if index<0:
uv=set(used_vars_set(strat))
lv=set([iter(p.lead()).next().index() for p in strat if p.lead_deg()==1])
candidates=uv.difference(lv)
if len(candidates)>0:
index=iter(candidates).next().index()
if index>=0:
print "chosen index:", index
step(strat, trace, Polynomial(Monomial(Variable(index))),0)
step(strat, trace, Polynomial(Monomial(Variable(index))),1)
else:
print "FINAL!!!", index
strat=symmGB_F2_python(strat, prot=True)
if not strat.containsOne():
print "TRACE", trace
print "SOLUTION"
for p in strat.minimalize_and_tail_reduce():
print p
raise Exception
def branch(strat, trace):
print "branching"
index = strat.suggestPluginVariable()
if index < 0:
uv = set(used_vars_set(strat))
lv = set([iter(p.lead()).next().index() for p in strat if p.
lead_deg() == 1])
candidates = uv.difference(lv)
if len(candidates) > 0:
index = iter(candidates).next().index()
if index >= 0:
print "chosen index:", index
step(strat, trace, Polynomial(Monomial(Variable(index))), 0)
step(strat, trace, Polynomial(Monomial(Variable(index))), 1)
else:
print "FINAL!!!", index
strat = symmGB_F2_python(strat, prot=True)
if not strat.containsOne():
print "TRACE", trace
print "SOLUTION"
for p in strat.minimalize_and_tail_reduce():
print p
raise Exception
def eliminate_ll_ranked(ll_system,
to_reduce,
reduction_function=ll_red_nf_noredsb,
reduce_ll_system=False,
prot=False):
assert (ll_system)
from_ring = ll_system[0].ring()
ll_ranks = rank(ll_system)
add_vars = set(used_vars_set(to_reduce).variables()).difference(
ll_ranks.keys())
for v in add_vars:
ll_ranks[v] = -1
#pushing variables ignored by ll to the front means,
#that the routines will quickly eliminate them
#and they won't give any overhead
def sort_key(v):
return (ll_ranks[v], v.index())
sorted_vars = sorted(ll_ranks.keys(), key=sort_key)
def var_index(v):
return iter(Monomial(v).variables()).next().index()
#sorted_var_indices=[var_index(v) for v in sorted_vars]
to_ring = Ring(len(sorted_vars))
map_back_indices = dict([(i, var_index(v))
for (i, v) in enumerate(sorted_vars)])
map_from_indices = dict([(var_index(v), i)
for (i, v) in enumerate(sorted_vars)])
#dict([(v,k) for (k,v) in enumerate(sorted_var_indices)])
var_names = [str(v) for v in sorted_vars]
try:
for (i, v) in enumerate(sorted_vars):
assert var_names[i] == str(v), (var_names[i], v, var_index(v), i)
# _set_variable_name(to_ring, i, var_names[i] + "TO")
finally:
pass
try:
map_from_vec = construct_map_by_indices(to_ring, map_from_indices)
finally:
pass
map_back_vec = construct_map_by_indices(from_ring, map_back_indices)
def map_from(p):
res = substitute_variables(to_ring, map_from_vec, p)
# assert str(p)==str(res), (str(p), str(res), list(map_from_vec),
# list(map_back_vec))
return res
def map_back(p):
return substitute_variables(from_ring, map_back_vec, p)
try:
ll_opt_encoded = ll_encode([map_from(p) for p in ll_system],
prot=False,
reduce=reduce_ll_system)
def llnf(p):
return map_back(reduction_function(map_from(p), ll_opt_encoded))
opt_eliminated = [llnf(p) for p in to_reduce]
finally:
pass
return (llnf, opt_eliminated)
], kwd_args)
manager = VariableManager(ring=ring,
include_outputs=options.include_outputs,
initialize=options.initialize,
**kwd_args)
from sys import argv
f = args[0]
parse(f, manager)
(ideal_state, ideal_intermediate, ideal_next_state) = manager.ideals()
ideal = ideal_state + ideal_intermediate + ideal_next_state
variables = []
used_vars = set(used_vars_set(ideal).variables())
if not options.forward:
variables = list(
chain(reversed(manager.next), reversed(manager.output),
reversed(manager.intermediate), reversed(manager.input),
reversed(manager.state)))
else:
variables = list(
chain(reversed(manager.output), reversed(manager.intermediate),
reversed(manager.input), reversed(manager.state),
reversed(manager.next)))
variables = ["t"] + variables
beginnings = [str(v)[:1] for v in variables]
block_starts = []
last = beginnings[0]
def evaluates_to_variables(self, signal):
ands = self.ands[signal]
return list(
used_vars_set(ands).variables())
def evaluates_to_variables(self, signal):
ands = self.ands[signal]
return list(used_vars_set(ands).variables())
def add_polynomial_to_cluster(self, p):
self.cluster.add(p)
self.used_variables_cluster = set(used_vars_set(self.cluster).variables())
self.number_of_used_variables_in_cluster = len(self.used_variables_cluster)
self.adjust_variables_introduction_mapping(self.used_variables_of_polynomial[p])
Block("xinter", inter_max, reverse=True),
Block("xinput", input_max, reverse=True),
Block("xstate", state_max, reverse=True)], kwd_args)
manager = VariableManager(ring=ring, include_outputs=options.
include_outputs, initialize=options.initialize, **kwd_args)
from sys import argv
f = args[0]
parse(f, manager)
(ideal_state, ideal_intermediate, ideal_next_state) = manager.ideals()
ideal = ideal_state + ideal_intermediate + ideal_next_state
variables = []
used_vars = set(used_vars_set(ideal).variables())
if not options.forward:
variables = list(chain(reversed(manager.next), reversed(manager.output
), reversed(manager.intermediate), reversed(manager.input),
reversed(manager.state)))
else:
variables = list(
chain(reversed(manager.output),
reversed(manager.intermediate),
reversed(manager.input),
reversed(manager.state),
reversed(manager.next)))
variables = ["t"] + variables
beginnings = [str(v)[:1] for v in variables]
block_starts = []
last = beginnings[0]
