Category: cryptography

def find_prime_near(x):
while True:
if is_prime(x):
return x
x = x + 1

Naive primality test
def is_prime(x):
for i in range(2, x):
if x % i == 0: return False
return True

Faster Primality Testing
probabalistic test
passes the test, p(x is composite) <= 2 -k Fermat's Little Theorem: if p is prime, 1 <= a < p => ap-1 = 1 mod p

from Crypto.Cipher import AES
from copy import copy

def find_collision(message):
	new_message = copy(message)

	return new message

def test():
	messages = ["Trust, but verify. -a signature phrase of President Ronald Reagan",
                "The best way to find out if you can trust somebody is to trust them. (Ernest Hemingway)",
                "If you reveal your secrets to the wind, you should not blame the wind for revealing them to the trees. (Khalil Gibran)",
                "I am not very good at keeping secrets at all! If you want your secret kept do not tell me! (Miley Cyrus)",
                "This message is exactly sixty four characters long and no longer"]
    for m in messages:
    	m = string_to_bits(m)
    	new_message = find_collision(m)
    	if not check(m, new_message):
    		print "Failed to find a collision for '%s'" % m
    		return False
    return True

from Crypto.Cipher import AES

BITS = ('0', '1')
ASCII_BITS = 8

def display_bits(b):
    """converts list of {0, 1}* to string"""
    return ''.join([BITS[e] for e in b])

def seq_to_bits(seq):
    return [0 if b == '0' else 1 for b in seq]

def pad_bits(bits, pad):
    """pads seq with leading 0s up to length pad"""
    assert len(bits) <= pad
    return &#91;0&#93; * (pad - len(bits)) + bits
        
def convert_to_bits(n):
    """converts an integer `n` to bit array"""
    result = &#91;&#93;
    if n == 0:
        return &#91;0&#93;
    while n > 0:
        result = [(n % 2)] + result
        n = n / 2
    return result

def string_to_bits(s):
    def chr_to_bit(c):
        return pad_bits(convert_to_bits(ord(c)), ASCII_BITS)
    return [b for group in 
            map(chr_to_bit, s)
            for b in group]

def bits_to_char(b):
    assert len(b) == ASCII_BITS
    value = 0
    for e in b:
        value = (value * 2) + e
    return chr(value)

def list_to_string(p):
    return ''.join(p)

def bits_to_string(b):
    return ''.join([bits_to_char(b[i:i + ASCII_BITS]) 
                    for i in range(0, len(b), ASCII_BITS)])

def pad_bits_append(small, size):
    # as mentioned in lecture, simply padding with
    # zeros is not a robust way way of padding
    # as there is no way of knowing the actual length
    # of the file, but this is good enough
    # for the purpose of this exercise
    diff = max(0, size - len(small))
    return small + [0] * diff

def xor_bits(bits_a, bits_b):
    """returns a new bit array that is the xor of `bits_a` and `bits_b`"""
    return [a^b for a, b in zip(bits_a, bits_b)]

def bits_inc(bits):
    """modifies `bits` array in place to increment by one
    
    wraps back to zero if `bits` is at its maximum value (each bit is 1)
    """
    # start at the least significant bit and work towards
    # the most significant bit
    for i in range(len(bits) - 1, -1, -1):
        if bits[i] == 0:
            bits[i] = 1
            break
        else:
            bits[i] = 0

def aes_encoder(block, key):
    block = pad_bits_append(block, len(key))
    # the pycrypto library expects the key and block in 8 bit ascii 
    # encoded strings so we have to convert from the bit array
    block = bits_to_string(block)
    key = bits_to_string(key)
    ecb = AES.new(key, AES.MODE_ECB)
    return string_to_bits(ecb.encrypt(block))

def get_block(plaintext, i, block_size):
    """returns the ith block of `plaintext`"""
    start = i * block_size
    if start >= len(plaintext):
        return None
    end = min(len(plaintext), (i+1) * block_size)
    return pad_bits_append(plaintext[start:end], block_size)

def get_blocks(plaintext, block_size):
    """iterates through the blocks of blocksize"""
    i = 0
    while True:
        start = i * block_size
        if start >= len(plaintext):
            break
        end = (i+1) * block_size
        i += 1
        yield pad_bits_append(plaintext[start:end], block_size)

def _counter_mode_inner(plaintext, key, ctr, block_enc):
    eblock = block_enc(ctr, key)
    cblock = xor_bits(eblock, plaintext)
    bits_inc(ctr)
    return cblock

def counter_mode(plaintext, key, ctr, block_size, block_enc):
    """Return the counter mode encoding of `plaintext"""
    cipher = []
    # break the plaintext into blocks
    # and encode each one
    for block in get_blocks(plaintext, block_size):
        cblock = _counter_mode_inner(block, key, ctr, block_enc)
        cipher.extend(cblock)
    return cipher

def counter_mode_hash(plaintext):
    block_size, block_enc, key, ctr = hash_inputs()
    hash_ = None
    for block in get_blocks(plaintext, block_size):
        cblock = _counter_mode_inner(block, key, ctr, block_enc)
        if hash_ is None:
            hash_ = cblock
        else:
            hash_ = xor_bits(hash_, cblock)
    return hash_

def hash_inputs():
    block_size = 128
    block_enc = aes_encoder
    key = string_to_bits("Vs7mHNk8e39%CXeY")
    ctr = [0] * block_size
    return block_size, block_enc, key, ctr

def _is_same(bits_a, bits_b):
    if len(bits_a) != len(bits_b):
        return False
    for a, b in zip(bits_a, bits_b):
        if a != b:
            return False
    return True

def check(message_a, message_b):
    """return True if `message_a` and `message_b` are
    different but hash to the same value"""

    if _is_same(message_a, message_b):
        return False
    
    hash_a = counter_mode_hash(message_a)
    hash_b = counter_mode_hash(message_b)

    return _is_same(hash_a, hash_b)

from Crypto.Cipher import AES

def non_encoder(block, key):
	"""A basic encoder that doesn't actually do anything"""
	return pad_bits_append(block, len(key))

def xor_encoder(block, key):
	block = pad_bits_append(block, len(block, len(key))
	cipher = [b ^ k for b, k in zip(block, key)]
	return cipher

def aes_encoder(block, key):
	block = pad_bits_append(block, len(key))
	block = bit_to_string(block)
	key = bits_to_string(key)
	ecb = AES.new(key, AES.MODE_ECB)
	return string_to_bits(ecb.encrypt(block))

def electronic_cookbook(plaintext, key, block_size, block_enc):
	"""Return the ecb encoding of 'plaintext"""
	cipher = []
	for i in range(len(plaintext)/ block_size + 1):
		start = i * block_size
		if start >= len(plaintext):
			break
		end = min(len(plaintext), (i+1) * block_size)
		block = plaintext[start:end]
		cipher.extend(block_enc(block, key))
	return cipher

def cipher_block_chaining(plaintext, key, init_vec, block_size, block_enc):
	"""Return the cbc encoding of 'plain text'"""

def test():
	key = string_to_bit('4h8f.093mJo:*9#$')
	iv = string_to_bit('89JIlkj3$%0lkjdg')
	plaintext = string_to_bits("one if by land; two if by sea")


BITS = ('0', '1')
ASCII_BITS = 8

def display_bits(b):
	"""converts list of {0, 1}* to string"""
	return ''.join([ITS[e] for e in b])

def seq_to_bits(seq):
	return [0 if b == '0' else 1  for b in seq]

def pad_bits(bits, pad):
	"""pads seq with leading 0s up to length pad"""
	assert len(bits) <= pad
	return &#91;0&#93; * (pad - len(bits)) + bits

def convert_to_bits(n):
	"""converts an integer 'n' to bit array"""
	result = &#91;&#93;
	if n == 0:
		return &#91;0&#93;
	while n > 0:
		result = [(n % 2)] + result
		n = n / 2
	return result

def string_to_bits(s):
	def chr_to_bit(c):
		return pad_bit(convert_to_bits(ord(c)), ASCII_BITS)
	return [b for group in
			map(chr_to_bits, s)
			for b in group]

def bits_to_char(b):
	assert len(b) == ASCII_BITS
	value = 0
	for e in b:
		value = (value * 2) + e
	return chr(value)

def list_to_string(p):
	return ''.join(p)

def bits_to_string(b):
	return ''.join([bits_to_char(b[i:i + ASCII_BITS])
				for i in range(0, len(b), ASCII_BITS)])

def pad_bits_append(small, size):
	diff = max(0, size - len(small))
	return small + [0] * diff

Quantum Mechanics
Thermal Noise
Key proceed mouse move

Peseudo-Random Number Generator
counter 0, 1, 2 -> Encrypt
Pool of Randomness seed-> key

Storing a file securely

Electronic Codebook Mode

def generate_sequence(f, n):
	return map(f, range(n))


def generate_fake_random(n):
	repetition = 0
	previous = None
	res = []
	for i in range(n):
	x = random.choice([0, 1])
	if x == previous:
		repetition += 1
		if repetition > 2:
			x = (x + 1) % 2
			repetition = 1
			previous = x
	else:
		previous = x
		repetition = 1
	res.append(x)
return res

length = 88

print display_bits(generate_sequence(lambda n: 0 if n % 3 == 0 else 1, length))

s i random if K(s) = |s| + c

BITS = ('0', '1')
ASCII_BITS = 7

def display_bits(b):
	"""converts list of {0, 1}* to string"""
	return ''.join([BITS[e] for e in b])

def seq_to_bits(seq):
	return [0 if b == '0' else 1 for b in seq]

def pad_bits(bits, pad):
	"""pads seq with leading 0s up to length pad"""
	assert len(bits) <= pad
	return &#91;0&#93; * (pad - len(bits)) + bits

def convert_to_bits(n):
	"""converts an integer 'n' to bit array"""
	result = &#91;&#93;
	if n == 0:
		return &#91;0&#93;
	while n > 0:
		result = [(n % 2)] + result
		n = n / 2
	return result

def string_to_bits(s):
	def chr_to_bit(c):
		return pad_bits(convert_to_bits(ord(c)), ASCII_BITS)
	return [b for group in
			map(chr_to_bit, s)
			for b  in group]

def bits_to_char(b):
	assert len(b) == ASCII_BITS
	value = 0
	for e in b:
		value = (value * 2) + e
	return chr(value)

def list_to_string(p):
	return ''.join(p)

def bits_to_string(b):
	return ''.join([bits_to_char(b[i:i + ASCII_BITS])
		for i in range(0, len(b), ASCII_BITS)])

Machine generates key sequence based on its configuration
key would not repeat for 10^19 letters

Event: subset of outcomes from a probability space
Head = { H }
Valid = { H, T }
P(A) = ΣweA P(w)

1 – P(E) = P(Valid)

Complementary Event Prbability
VA: P(a) + p(A) = 1

Conditional probability
P(B|A) = P(Aub)/ p(A)

Perfect Cipher
the ciphertext provides an attacker with no additional information about the plaintext.
m -> enc -> c
m e M
m* e M

Eki(mi) = c
P(m=m*).P(k=k*) = 1/k

P(Ek(m)= c) = 1/|k|
P(m=m* Ek(m)=c) = P(m = m*)/|k|

Malleable
alice Ek(m)=c active attacker

Impratical |k| = |m|

Shamon’s Theorem: if a cipher is perfect, it must be impractical

Perfect cipher: P(m=m*|Ek(m)=c) = P(m=m*)

Ek(m) = c0 c1 … Cn-1
m = ‘CS’ = 1010011 1000011
k = 11001000100110
c = 01101111100101

Proving Security
here is a mathematical proof, accepted by experts, that shows the cipher is secure.

Probability(Review)
Ω:set of all possible outcomes
probability space
Ω = {h, t}
uniform distribution each outcome is equally likely
p: outcome -> [0, 1]
p(h) = 1/2 p(t) = 1/2
Ω = {H, T, E}
p(h) = 0.49999
p(t) = 0.49999
ΣweΩ　p(w) = 1

plaintext -> encryption -> ciphertext -> insecure channel -> decryptron
VmeM:D(E(m))=m
Kerck hoff’s Principle CIS

meM ceC keK
E:MxK -> c D:c*k -> M
correctness property: Vm;k: Dk(Ek(m))

Security property:
Ideal: ciphertext reveals nothing about key or message

def convert_to_bits(n, pad):
	result = []
	while n > 0:
		if n % 2 == 0:
			result = [0] + result
		else:
			result = [1] + result
		n = n / 2
	while len(result) < pad:
		result = [0] + result
	return result