def symbolic_moo_gen(session_label, block_label):
c = Function("C", 3)
p = Constant(session_label)
i = Constant(block_label)
a = Constant("1")
pList = [
Variable(f"x{session_label}{block_label}"),
Variable(f"x{session_label}{block_label}"),
Variable(f"x{session_label}{block_label}")
]
cList = [c(p, i, a), c(p, i, a), c(p, i, a)]
return program.chaining_function(3, [0], pList, cList)
def program():
"""
Webpage that walks the user through an interaction between
the adversary and the oracle.
"""
if request.method != "POST":
# If you step away from the simulation, destroy the previous session
if 'uid' in session and session['uid'] in moo_sessions.keys():
del moo_sessions[session['uid']]
session.pop('uid', None)
return render_moo_template('program_create.html')
# If a MOOProgram is already in session....
if 'uid' in session and session['uid'] in moo_sessions.keys():
uid = session['uid']
moo_session: MOOProgram = moo_sessions[uid]
if 'next' in request.form:
plaintext = Variable("x_" + str(moo_session.iteration))
ciphertext = moo_session.rcv_block(plaintext)
response = str(
ciphertext
) if ciphertext is not None else "Sent " + str(plaintext)
return render_moo_template('program.html', response=response)
if 'end' in request.form:
ciphertext = moo_session.rcv_stop()
del moo_sessions[uid]
session.pop('uid', None)
response = str(ciphertext) if ciphertext is not None else None
if response is None:
return render_moo_template('program_create.html')
return render_moo_template('program.html', response=response)
# If we are creating a session of a MOOProgram....
if 'chaining' in request.form and 'schedule' in request.form:
# Create new session
chaining_moo = request.form.get('chaining')
schedule = request.form.get('schedule')
moo_session = MOOProgram(chaining_moo, schedule)
# Send an initial message
plaintext = Variable("x_" + str(moo_session.iteration))
ciphertext = moo_session.rcv_block(plaintext)
# Set up a userid and save the moo_session
session['uid'] = uuid4()
moo_sessions[session['uid']] = moo_session
response = str(
ciphertext) if ciphertext is not None else "Sent " + str(plaintext)
return render_moo_template('program.html', response=response)
return render_moo_template('program_create.html')
def _temporary_parser(moo_string: str) -> Term:
"""
A temporary parser to parse a user
supplied cryptographic mode of operation.
This function is limited in what it can parse.
"""
parser = Parser()
parser.add(Function("f", 1))
parser.add(xor)
parser.add(Variable("P[i]"))
parser.add(Variable("C[i]"))
parser.add(Variable("C[i-1]"))
parser.add(Constant("r"))
parser.add(Constant("P[0]"))
return parser.parse(moo_string)
def test_pick_fail():
c = Function("C", 3)
f = Function("f", 1)
a = Constant("a")
b = Constant('1')
e = Constant("e")
p = Constant('p')
q = Constant('q')
i = Constant('i')
j = Constant('j')
x = Variable("x")
z = Zero()
func = FuncTerm(f, [a])
func2 = FuncTerm(f, [e])
func_pi = FuncTerm(c, [p, i, b])
func_qj = FuncTerm(c, [q, j, b])
eq1 = Equation(func, z)
eq2 = Equation(func_qj, x)
eq3 = Equation(xor(func_pi, func), z)
eq4 = Equation(xor(xor(func_pi, func), func2), z)
topf: Set[Equation] = {eq1, eq4}
print("Testing pick_fail with ", topf)
new_set = pick_fail(topf, cbc_gen)
print("Result: ", new_set)
print()
topf: Set[Equation] = {eq1, eq2, eq3}
print("Testing pick_fail with ", topf)
new_set = pick_fail(topf, ex4_gen)
print("Result: ", new_set)
print()
def test_check_xor_structure():
f = Function("f", 1)
a = Constant("a")
b = Constant("b")
c = Constant("c")
x = Variable("x")
z = Zero()
func = FuncTerm(f, [a])
func1 = FuncTerm(f, [b])
func2 = FuncTerm(f, [c])
func3 = FuncTerm(f, [x])
func4 = xor(func, func1)
func5 = xor(func2, func3)
eq1 = Equation(func, b)
eq2 = Equation(func3, c)
eq3 = Equation(func5, z)
eq4 = Equation(xor(func2, func3), z)
eq5 = Equation(xor(func4, func5), z)
topf: Set[Equation] = {eq1, eq2, eq3, eq5}
print("Testing pick_f with equation set: ", topf)
print("Result pick_f: ", pick_f(topf))
print()
def ex4_gen(p, i):
if i == 0:
return Constant('r' + str(p))
else:
return xor(
xor(FuncTerm(f, [ex4_gen(p, i - 1)]),
FuncTerm(f, [FuncTerm(f, [ex4_gen(p, i - 1)])])),
Variable("x" + str(p) + str(i)))
def _satisfies_chaining(term: Term, max_history: int):
"""Checks to see if a preivous ciphertext is in the term."""
previous_ciphertexts = [
Variable("C[i-" + str(i + 1) + "]") for i in range(max_history)
]
previous_ciphertexts_in_term = [(ci in term)
for ci in previous_ciphertexts]
return any(previous_ciphertexts_in_term)
def symbolic_cbc_gen(session_label, block_label):
a = Constant("1")
p = Constant(session_label)
i = Constant(block_label)
cInner = FuncTerm(c, [p, i, a])
x = Variable("xpi")
return xor(FuncTerm(f, [cInner]), x)
def invert_simple(term):
"""
Algorithm to get the plaintext P_{i} out of a MOO.
Works by moving up in the DAG from the plaintext variable
Applying the reverse transformation as it moves along
f -> f^{-1}
xor(P, x) -> xor(P, xor(x, x)) -> P
Assumptions:
- Previous plaintexts aren't in the term
- The plaintext is only in one position
"""
inverse_term = Variable("x")
transforms = []
# Create a TermDAG and check to see
# that the plaintext is only in one position.
d = TermDAG(term)
assert Counter(d.leaves())[_P] == 1
# Move up the DAG starting from the plaintext
# leaf and add inverse operations to the
# transforms list.
current_term = deepcopy(_P)
parent_term = list(d.parents(current_term))
while parent_term != []:
parent_term = parent_term[0]
if parent_term.function == _f:
transforms.append((_finv, ))
elif parent_term.function == xor:
# Cancel out the xor by xoring with
# the other argument again.
if parent_term.arguments[0] == current_term:
transforms.append((xor, parent_term.arguments[1]))
else:
transforms.append((xor, parent_term.arguments[0]))
else:
raise ValueError("A function other than f or xor detected")
current_term = deepcopy(parent_term)
parent_term = list(d.parents(parent_term))
# Construct inverse term
for transform in reversed(transforms):
if transform[0] == _finv:
inverse_term = _finv(inverse_term)
else: # Assume xor
inverse_term = xor(inverse_term, transform[1])
return inverse_term
def test_elimf():
f = Function("f", 1)
xo = Function("f", 2)
z = Zero()
c = Constant("c")
x = Variable("x")
b = Variable("b")
func = FuncTerm(f, [x])
func2 = FuncTerm(f, [z])
func3 = FuncTerm(xo, [c, b])
eq1 = Equation(func, c)
eq2 = Equation(xor(func, func3), z)
eq3 = Equation(b, func)
eq4 = Equation(func2, z)
topf: Set[Equation] = {eq1, eq2, eq3, eq4}
print("Testing elim_f with ", topf)
new_set = elim_f(topf)
print("Result ", new_set)
print()
def symbolic_ex4_gen(session_label, block_label):
a = Constant("1")
p = Constant(session_label)
i = Constant(block_label)
cInner = FuncTerm(c, [p, i, a])
x = Variable("xpi")
fSummandOne = FuncTerm(f, [cInner])
fSummandTwo = FuncTerm(f, [FuncTerm(f, [cInner])])
return xor(fSummandOne, fSummandTwo, x)
def test_occurs():
f = Function("f", 1)
g = Function("g", 1)
c = Function("C", 3)
p = Constant('p')
q = Constant('q')
i = Constant('i')
z = Variable("z")
x = Variable("x")
y = Variable("y")
b = Variable("b")
a = Variable("a")
cpi = FuncTerm(c, [p, i, Constant("1")])
fcpi = FuncTerm(f, [cpi])
e1 = Equation(cpi, fcpi)
e2 = Equation(x, FuncTerm(f, [g(x)]))
e3 = Equation(x, FuncTerm(f, [b]))
occ = {e1, e2, e3}
print("Testing occurs check with ", occ)
print("new set: ", occurs_check(occ))
print()
def _changeVars(overlaping_vars: List[Variable], term: Term, hypothesis: Term,
conclusion: Term):
"""Change variable names in hypothesis & conclusion to not overlap with overlapping_vars"""
hypothesis = deepcopy(hypothesis)
conclusion = deepcopy(conclusion)
all_vars = get_vars(term, unique=True) | \
get_vars(hypothesis, unique=True) | \
get_vars(conclusion, unique=True)
new_vars: List[Variable] = []
# Go through all the variables that share the same symbol between the term and rewrite rule
# and change the variables in the rewrite rule
for v in overlaping_vars:
new_var = v
# Keep renaming variable in rewrite rule until it is not an already existing variable
while new_var in all_vars:
new_var = Variable(new_var.symbol + "_1", new_var.sort)
new_vars.append(new_var)
# Create substitution between the old and new variable names and apply them
s = SubstituteTerm()
for old_v, new_v in zip(overlaping_vars, new_vars):
s.add(old_v, new_v)
return hypothesis * s, conclusion * s
def InvertMOO(term: Term, plaintext: str, nonces: list, nonceone: Constant,
knowsiv: bool):
"""
NOT COMPLETE
Implementation of Lemma 10 from the Indocrypt paper
Parameters
==========
Term:
The first recursive case
str:
The plaintext to check for
bool:
Is the IV known
Examples
========
from symcollab.algebra import Constant, Variable
from symcollab.moe.program import MOOProgram
from symcollab.moe.check import moo_check
from symcollab.Unification.constrained.xor_rooted_unif import XOR_rooted_security
from symcollab.Unification.constrained.p_unif import p_unif
result = moo_check('cipher_block_chaining', "every", p_unif, 2, True, True)
print(result.invert_result)
"""
if nonceone in get_constants(term, unique=True):
if knowsiv == False:
return False
#Make sure C_0 is only IV
if len(nonces) == 1:
plaintext = Variable(plaintext)
# Make sure C_i contains p_i but only once
if count_occurence(plaintext, term) == 1:
return True
# Passes all the criteria
return False
def rcv_block(self, message: Term) -> Optional[MOOFrame]:
"""
The adversary gives a plaintext block to the oracle to encrypt.
Depending on the schedule, the oracle might return back a MOOFrame.
"""
if self.stopped:
raise ValueError("Session already stopped.")
self.iteration += 1
self.plain_texts.append(message)
# Create new cipher text variable and map it to the MOO Term
new_cipher_var = Variable(f"y_{self.iteration}")
encrypted_block = self.chaining_function(self.iteration, self.nonces,
self.plain_texts,
self.cipher_texts)
self.substitutions.add(new_cipher_var, encrypted_block)
self.cipher_texts.append(new_cipher_var)
# If the schedule allows, return a MOO Frame
if self.schedule(self.iteration):
return MOOFrame(deepcopy(new_cipher_var),
deepcopy(self.substitutions))
return None
def test_pick_c():
c = Function("C", 3)
f = Function("f", 1)
a = Constant("a")
b = Constant('1')
p = Constant('p')
q = Constant('q')
i = Constant('i')
j = Constant('j')
x = Variable("x")
z = Zero()
func = FuncTerm(f, [a])
func_pi = FuncTerm(c, [p, i, b])
func_qj = FuncTerm(c, [q, j, b])
eq1 = Equation(func, z)
eq2 = Equation(func_qj, x)
eq3 = Equation(xor(func_pi, func), 0)
topf: Set[Equation] = {eq1, eq2, eq3}
print("Testing pick_c with ", topf)
def test_elimc():
c = Function("C", 3)
p = Constant('p')
q = Constant('q')
i = Constant('i')
j = Constant('j')
a = Constant('1')
x = Variable("x")
z = Zero()
func_pi = FuncTerm(c, [p, i, a])
func_qj = FuncTerm(c, [q, j, a])
eq1 = Equation(xor(func_pi, func_qj), z)
eq2 = Equation(xor(func_qj, func_pi), z)
eq3 = Equation(func_pi, x)
topf: Set[Equation] = {eq1, eq2, eq3}
print("Testing elim_c with ", topf)
new_set = elim_c(topf)
print("Result: ", new_set)
print()
b = Constant('2')
func_pi = FuncTerm(c, [p, i, a])
func_qj = FuncTerm(c, [q, j, b])
eq1 = Equation(xor(func_pi, func_qj), z)
eq2 = Equation(xor(func_qj, func_pi), z)
topf: Set[Equation] = {eq1, eq2}
print("Testing elim_c with ", topf)
new_set = elim_c(topf)
print("Result: ", new_set)
print()
def name_xor_terms(t, eqs):
# Purify arguments
# Name top-level xor-subterms
# Return (renamed_term, a set of equations)
if is_zero(t):
return (t, eqs)
elif (isinstance(t, Constant)):
return (t, eqs)
elif (isinstance(t, Variable)):
return (t, eqs)
elif (is_xor_term(t)):
(term1, eqs) = purify_a_term(t.arguments[0], eqs)
#eqs = eqs + equation1
(term2, eqs) = purify_a_term(t.arguments[1], eqs)
#eqs = eqs + equation2
#equations = equation1 + equation2
name = look_up_a_name(xor(term1, term2), eqs)
if (name != None):
return (name, eqs)
else:
global variable_counter
variable_counter += 1
new_variable = Variable("N" + str(variable_counter))
eqs.append(Equation(new_variable, xor(term1, term2)))
return (new_variable, eqs)
elif (isinstance(t, FuncTerm)):
terms = []
for arg in t.arguments:
(term, eqs) = name_xor_terms(arg, eqs)
#eqs = eqs + equations
terms.append(term)
return (FuncTerm(t.function, terms), eqs)
else:
print("error")
return None
#!/usr/bin/env python3
from symcollab.algebra import Function, Variable, Constant, Equation
from symcollab.Unification.unif import unif
# Setting up terms
f = Function("f", 2)
g = Function("g", 1)
x = Variable("x")
y = Variable("y")
z = Variable("z")
a = Constant("a")
b = Constant("b")
# applying unification
#example 1: unifiable
unif({Equation(f(x, y), f(a, b))})
#example 2: simple function clash
unif({Equation(f(x, y), g(z))})
#example 3: function clash
unif({Equation(f(x, x), f(g(y), a))})
#example 4: occurs check
unif({Equation(f(x, y), f(g(x), a))})
#example 5: unifiable
unif({Equation(f(z, z), f(g(f(x, y)), g(f(a, b))))})
#!/usr/bin/env python3
from symcollab.algebra import Constant, Function, Variable
from symcollab.rewrite import RewriteRule
a = Constant("a")
b = Constant("b")
c = Constant("c")
x = Variable("x")
y = Variable("y")
f = Function("f", 2)
g = Function("g", 2)
r = RewriteRule(f(y, g(x, a)), g(y, a))
term = f(b, g(c, a))
print("Applying " + str(r) + " to " + str(term))
print("Result:", r.apply(term))
print("Now to show what happens when you can't apply a term...")
term = f(a, b)
print("Applying " + str(r) + " to " + str(term))
print("Result:", r.apply(term))
print("Applying f(x, x) -> x to f(f(x, x), f(x, x))")
term = f(f(x, x), f(x, x))
r = RewriteRule(f(x, x), x)
print("Result:", r.apply(term))
print("Applying f(x, x) -> x to f(f(x, x), f(x,x)) at position 2")
print("Result:", r.apply(term, '2'))
def Launcher():
window = make_window()
window.finalize()
filename = "moo_output.txt"
f_writetype = "w+"
fileinfo = ""
moo_session = None
window_size = window.Size
# event loop
while True:
event, values = window.read() # type: str, dict
# if the window is closed leave loop immediately
if event == sg.WIN_CLOSED:
break
#print(event, values)
start, stop = 1, 6
result = []
popup = True
goodInput = True
function = ''
'''Commenting this out because not sure if it is
desired or not
if event == 'Submit settings':
filename = str(values[22]) + ".txt"
if values[23] == 'Append':
f_writetype = "a+"'''
# all tool tab events
if event == 'Execute!':
# check all input
start, stop = 1, 6
function = 'tool'
for i in range(start, stop + 1):
result.append(values[i])
sim_next = False
# all simulation tab events
if event == 'Execute!0' or event == 'Next>>':
if event == 'Next>>':
sim_next = True
# check all input
start, stop = 7, 8
function = 'simulation'
for i in range(start, stop):
result.append(values[i])
result.append(values[stop])
# all custom tab events
if event == 'Execute!1':
# check all input
start, stop = 9, 14
function = 'custom'
for i in range(start, stop + 1):
result.append(values[i])
# all random tab events
if event == 'Execute!2':
#check all input
start, stop = 15, 24
function = 'random'
for i in range(start, stop + 1):
result.append(values[i])
# check if any of the inputs are blank
if all(result) is not True:
sg.Popup('Input left blank, please enter a value.', title='ERROR')
popup = False
goodInput = False
if function == 'tool' and goodInput:
if valid_moo_unif_pair(result[0], result[1]) == False:
goodInput = False
sg.Popup(
'Invalid unfication algorithm and chaining function combination, see Help -> Tool for more information',
title='ERROR')
# if none of the input is blank / combinations are good
#then perform tool functions and output results to window
if goodInput:
if function == 'tool':
fileinfo += "#######################################################################################################\n"
fileinfo += "Tool inputs: " + str(result) + "\n"
unif = unif_dict[result[0]]
chaining = cf_dict[result[1]]
sched = scd_dict[result[2]]
length_bound = restrict_to_range(int(result[3]), 0, 100)
knows_iv = yn_tf(result[4])
invert_check = yn_tf(result[5])
# check for security and catch exceptions
try:
result = moo_check(chaining, sched, unif, length_bound,
knows_iv, invert_check)
except ValueError as v_err:
message = 'ValueError: ' + str(v_err)
sg.Popup(message, title='VALUE ERROR')
window.close()
except:
print("Unexpected error:", sys.exc_info()[0])
window.close()
response = get_response(result)
fileinfo += "tool output:\n" + response + "\n"
window['-O1-'].update(response)
if function == 'simulation':
fileinfo += "#######################################################################################################\n"
fileinfo += "Simulation inputs: " + str(result) + "\n"
chaining = cf_dict[result[0]]
sched = scd_dict[result[1]]
if sim_next: # continuing session
if moo_session is None:
window['-O2-'].update("Begin session first!")
else:
plaintext = Variable("x_" + str(moo_session.iteration))
ciphertext = moo_session.rcv_block(plaintext)
response = str(
ciphertext
) if ciphertext is not None else "Sent " + str(
plaintext)
fileinfo += "Next: " + response + "\n"
window['-O2-'].update(response)
else: # creating the session
moo_session = MOOProgram(chaining, sched)
plaintext = Variable("x_" + str(moo_session.iteration))
ciphertext = moo_session.rcv_block(plaintext)
response = str(
ciphertext
) if ciphertext is not None else "Sent " + str(plaintext)
fileinfo += "Simulation begin:\n" + response + "\n"
window['-O2-'].update(response)
if function == 'custom':
fileinfo += "#######################################################################################################\n"
fileinfo += "Custom inputs: " + str(result) + "\n"
chaining = CustomMOO(_temporary_parser(result[0])).name
unif = unif_dict[result[1]]
sched = scd_dict[result[2]]
length_bound = restrict_to_range(int(result[3]), 0, 100)
knows_iv = yn_tf(result[4])
invert_check = yn_tf(result[5])
try:
result = moo_check(chaining, sched, unif, length_bound,
knows_iv, invert_check)
except ValueError as v_err:
message = 'ValueError: ' + str(v_err)
sg.Popup(message, title='VALUE ERROR')
window.close()
except:
print("Unexpected error:", sys.exc_info()[0])
window.close()
response = get_response(result)
fileinfo += "Custom output:\n" + response + "\n"
window['-O3-'].update(response)
if function == 'random':
fileinfo += "#######################################################################################################\n"
fileinfo += "Random inputs: " + str(result) + "\n"
unif = unif_dict[result[0]]
sched = scd_dict[result[1]]
length_bound = restrict_to_range(int(result[2]), 0, 100)
f_bound = int(result[3])
moo_bound = restrict_to_range(int(result[4]), 1, 100)
c_req = yn_tf(result[5])
iv_req = yn_tf(result[6])
sec_req = yn_tf(result[7])
knows_iv = yn_tf(result[8])
invert_check = yn_tf(result[9])
# generate random moos
filtered_gen = FilteredMOOGenerator(1, f_bound, iv_req, c_req)
moo_list = (next(filtered_gen) for i in range(moo_bound))
response = ""
# Check security of the modes of operation
moo_safe_list: List[Term] = list()
for random_moo_term in moo_list:
response += str(random_moo_term) + "\n"
cm = CustomMOO(random_moo_term)
moo_result = moo_check(cm.name, sched, unif, length_bound,
knows_iv, invert_check)
if moo_result.secure:
moo_safe_list.append(random_moo_term)
fileinfo += "Random output:\n" + response + "\n"
window['-O4-'].update(response)
# menu button events create popups
# my intent with these is for this to be a place to give users more information
# about how to use each part of the tool, but any of this can be removed as desired
if event == 'Tool':
sg.Popup('Valid unification algorithm and chaining function pairs:\n\n'
'Syntactic: Cipher Block Chaining, Propogating Cipher Block Chaining,' \
' Hash Cipher Block Chaining, Cipher Feedback, Output Feedback\n'
'F-Rooted P-XOR: Cipher Block Chaining, Propogating Cipher Block Chaining, Hash Cipher Block Chaining\n'
'XOR-Rooted P-XOR: Cipher Feedback, Output Feedback', title='Tool usage', line_width=200)
if event == 'Simulation':
sg.Popup('This is a blurb about how to use the simulation page',
title='Simulation usage')
if event == 'Custom':
sg.Popup('This is a blurb about how to use the custom page',
title='Custom usage')
if event == 'Random':
sg.Popup('This is a blurb about how to use the random page',
title='Random usage')
f = open(filename, f_writetype)
f.write(fileinfo)
f.close()
window.close()
def instantiate_a_constraint(constraint, sigma):
#Instantiate a constraint using sigma
#For example, if constraint is {x1: [a, b], x2: [f(x1)]}, and sigma is {x1 |-> a}
#then the result is {x1: [a, b], x2: [f(a)]}
new_constraints = {}
for key in constraint.keys():
old_constraint = constraint[key]
new_constraint = []
for o in old_constraint:
new_constraint.append(
ConstrainedTerm(o.term * sigma, SubstituteTerm()))
new_constraints.update({key: new_constraint})
return new_constraints
x1 = Variable("x1")
x2 = Variable("x2")
x3 = Variable("x3")
x4 = Variable("x4")
a = Constant("a")
b = Constant("b")
c = Constant("c")
d = Constant("d")
f = Function("f", 1)
t0 = a
t1 = xor(a, b)
t2 = xor(b, c)
t3 = f(xor(x1, c))
t4 = f(x2)
"""Definition and properties for two-arity tuple."""
from symcollab.algebra import Function, Variable
from symcollab.rewrite import RewriteRule, RewriteSystem
from .inductive import TheorySystem, Inductive
@Inductive
class Pair(TheorySystem):
pair = Function("pair", 2)
# Fix domain of pair to take anything
Pair.pair.domain_sort = None
# Variables for later rules
_a = Variable("a", sort=Pair.sort)
_b = Variable("b", sort=Pair.sort)
fst = Function("fst", 1, domain_sort=Pair.sort)
Pair.define(fst, RewriteSystem({
RewriteRule(fst(Pair.pair(_a, _b)), _a),
}))
lst = Function("lst", 1, domain_sort=Pair.sort)
Pair.define(lst, RewriteSystem({
RewriteRule(lst(Pair.pair(_a, _b)), _b),
}))
@classmethod
def from_bool(cls, x: bool) -> Term:
"""Converts a bool to a Boolean."""
return deepcopy(Boolean.trueb if x else Boolean.falseb)
@classmethod
def to_bool(cls, x: Term) -> bool:
"""Converts a Boolean to an bool."""
if not isinstance(x, FuncTerm) or x != Boolean.trueb or x != Boolean.falseb:
raise ValueError("to_bool function expects a simplified Boolean.")
return x == Boolean.trueb
# Variables for later rules
_n = Variable("n", sort=Boolean.sort)
_m = Variable("m", sort=Boolean.sort)
# Negation
neg = Function("neg", 1, domain_sort=Boolean.sort, range_sort=Boolean.sort)
Boolean.define(
neg,
RewriteSystem({
RewriteRule(neg(Boolean.trueb), Boolean.falseb),
RewriteRule(neg(Boolean.falseb), Boolean.trueb)
})
)
# Boolean And
andb = Function("andb", 2, domain_sort=Boolean.sort, range_sort=Boolean.sort)
Boolean.define(
#Example of using the deducible function
from symcollab.algebra import Function, Variable, Constant
from symcollab.moe.invertibility import deducible
x = Constant("x")
f = Function("f", 2)
p = Constant("p_i")
deducible(f(x, p), {x})
deducible(f(x, p), {})
#example of using invertibility with MOO security check
from symcollab.algebra import Constant, Variable
from symcollab.moe.program import MOOProgram
from symcollab.moe.check import moo_check
from symcollab.Unification.constrained.xor_rooted_unif import XOR_rooted_security
from symcollab.Unification.constrained.p_unif import p_unif
result = moo_check('cipher_block_chaining', "every", p_unif, 2, True, True)
print(result.invert_result)
#example of using invertibility by itself, not how it's intended to be used
#but can be done for testing
from symcollab.moe.invertibility import InvertMOO
from symcollab.xor import xor
f = Function("f", 1)
x = Variable("x")
IV = Constant("IV")
C1 = xor(x, IV)
print("MOO Invertible?", InvertMOO(C1, "x", [IV], IV, True))
it does not simplify to a normal form of a term.
Inspired by "Implementing a Propositional Logic Theorem Prover in Haskell"
by Laurence Edward Day
"""
from symcollab.algebra import Function, Variable
from symcollab.rewrite import RewriteRule, RewriteSystem
from .inductive import Inductive, TheorySystem
@Inductive
class Prop(TheorySystem):
pass
A = Variable("A", sort=Prop.sort)
B = Variable("B", sort=Prop.sort)
C = Variable("C", sort=Prop.sort)
Not = Function("not", 1, domain_sort=Prop.sort, range_sort=Prop.sort)
Prop.define(Not, RewriteSystem({RewriteRule(Not(Not(A)), A)}))
And = Function("and", 2, domain_sort=Prop.sort, range_sort=Prop.sort)
Prop.define(And, RewriteSystem({
RewriteRule(And(A, A), A),
}))
Or = Function("or", 2, domain_sort=Prop.sort, range_sort=Prop.sort)
Prop.define(Or, RewriteSystem({
RewriteRule(Or(A, A), A),
}))
#eac_unif_ex.py
from symcollab.algebra import Function, Equation, Variable
from symcollab.Unification.eac_unif import eac_unif
exp = Function("exp", 2)
x = Variable("x")
y = Variable("y")
w = Variable("w")
t = exp(x, y)
t2 = exp(x, w)
e1 = Equation(t, t2)
U = set()
U.add(e1)
z = Variable("z")
g = Function("g", 2)
t3 = g(z, z)
t4 = g(w, w)
e2 = Equation(t3, t4)
U.add(e2)
eac_unif(U)
u = Variable("u")
v = Variable("v")
t3 = exp(v, w)
t4 = g(x, y)
e3 = Equation(u, t3)
e4 = Equation(u, t4)
U = set()
U.add(e3)
U.add(e4)
eac_unif(U)
# TODO: Capture the commutative property of addition
@Inductive
class Ring(TheorySystem):
"""
A ring is an abelian group with another binary
operation that is associative, distributive over the
abelian operation, and has an ientity element.
"""
add = Function("add", 2)
mul = Function("mul", 2)
negate = Function("neg", 1)
zero = Constant("0")
_x = Variable("x", sort=Group.sort)
_y = Variable("y", sort=Group.sort)
_z = Variable("z", sort=Group.sort)
# TODO: Make Group.rules.rules match what the function symbols are
Ring.rules = RewriteSystem(Group.rules.rules | {
## Associativity Rules
# (x * y) * z → x * (y * z)
RewriteRule(Ring.mul(Ring.mul(_x, _y), _z), Ring.mul(_x, Ring.mul(_y, _z))),
## Zero rules
# 0 * x → 0
RewriteRule(Ring.mul(Ring.zero, _x), Ring.zero),
# x * 0 → 0
RewriteRule(Ring.mul(_x, Ring.zero), Ring.zero),
## Distributivity rules
# x * (y + z) → (x * y) + (x * z)
RewriteRule(Ring.mul(_x, Ring.add(_y, _z)), Ring.add(Ring.mul(_x, _y), Ring.mul(_x, _z))),
#!/usr/bin/env python3
from symcollab.algebra import Function, Variable, Constant
from symcollab.xor.xorhelper import *
from symcollab.xor.structure import *
f = Function("f", 1)
x = Variable("x")
y = Variable("y")
z = Variable("z")
x1 = Variable("x1")
x2 = Variable("x2")
x3 = Variable("x3")
a = Constant("a")
b = Constant("b")
c = Constant("c")
d = Constant("d")
ze = Zero()
t1 = xor(x, f(y), f(x1))
t2 = xor(y, f(z), f(x2))
t3 = xor(z, f(x), f(x3))
eq1 = Equation(ze, t1)
eq2 = Equation(ze, t2)
eq3 = Equation(ze, t3)
equations = Equations([eq1, eq2, eq3])
#t1 = f(xor(x, y))
#t2 = x
#eq1 = Equation(t1, t2)
#equations = Equations([eq1])
#!/usr/bin/env python3
from symcollab.algebra import Constant, Function, Variable, Equation
from symcollab.Unification.syntactic_ac_unification import *
#Setup the variables and AC function
f = Function("f", 2)
x = Variable("x")
y = Variable("y")
z = Variable("z")
a = Constant("a")
b = Constant("b")
x1 = Variable("x1")
y1 = Variable("y1")
z1 = Variable("z1")
#Example 1
e = Equation(f(x, y), f(x1, y1))
U = {e}
sol = synt_ac_unif(U) #For a single solution
# also sol = synt_ac)unif(U, True)
sol = synt_ac_unif(U, False) # For all solutions
#Example 2
e = Equation(f(x, x), f(y, y))
U = {e}
sol = synt_ac_unif(U)
#Example 3
e2 = Equation(f(x, x), f(f(y, y), f(z, z)))
U1 = {e2}
sol = synt_ac_unif(U1)
