r"""
Simplified DES
A simplified variant of the Data Encryption Standard (DES). Note that
Simplified DES or S-DES is for educational purposes only. It is a
small-scale version of the DES designed to help beginners understand the
basic structure of DES.
AUTHORS:
- Minh Van Nguyen (2009-06): initial version
"""
from sage.monoids.string_monoid import BinaryStrings
from sage.structure.sage_object import SageObject
class SimplifiedDES(SageObject):
r"""
This class implements the Simplified Data Encryption Standard (S-DES)
described in [Sch96]_. Schaefer's S-DES is for educational purposes
only and is not secure for practical purposes. S-DES is a version of
the DES with all parameters significantly reduced, but at the same time
preserving the structure of DES. The goal of S-DES is to allow a
beginner to understand the structure of DES, thus laying a foundation
for a thorough study of DES. Its goal is as a teaching tool in the same
spirit as Phan's
:mod:`Mini-AES <sage.crypto.block_cipher.miniaes>` [Pha02]_.
EXAMPLES:
Encrypt a random block of 8-bit plaintext using a random key, decrypt
the ciphertext, and compare the result with the original plaintext::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES(); sdes
Simplified DES block cipher with 10-bit keys
sage: bin = BinaryStrings()
sage: P = [bin(str(randint(0, 1))) for i in xrange(8)]
sage: K = sdes.random_key()
sage: C = sdes.encrypt(P, K)
sage: plaintxt = sdes.decrypt(C, K)
sage: plaintxt == P
True
We can also encrypt binary strings that are larger than 8 bits in length.
However, the number of bits in that binary string must be positive
and a multiple of 8::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: bin = BinaryStrings()
sage: P = bin.encoding("Encrypt this using S-DES!")
sage: Mod(len(P), 8) == 0
True
sage: K = sdes.list_to_string(sdes.random_key())
sage: C = sdes(P, K, algorithm="encrypt")
sage: plaintxt = sdes(C, K, algorithm="decrypt")
sage: plaintxt == P
True
REFERENCES:
.. [Pha02] R. C.-W. Phan. Mini advanced encryption standard (mini-AES): a
testbed for cryptanalysis students. Cryptologia, 26(4):283--306, 2002.
.. [Sch96] E. Schaefer. A simplified data encryption algorithm.
Cryptologia, 20(1):77--84, 1996.
"""
def __init__(self):
r"""
A simplified variant of the Data Encryption Standard (DES).
EXAMPLES::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES(); sdes
Simplified DES block cipher with 10-bit keys
sage: B = BinaryStrings()
sage: P = [B(str(randint(0, 1))) for i in xrange(8)]
sage: K = sdes.random_key()
sage: C = sdes.encrypt(P, K)
sage: plaintxt = sdes.decrypt(C, K)
sage: plaintxt == P
True
"""
from sage.crypto.mq import SBox
self._key_size = 10
self._sbox0 = SBox(1, 0, 3, 2, 3, 2, 1, 0, 0, 2, 1, 3, 3, 1, 3, 2)
self._sbox1 = SBox(0, 1, 2, 3, 2, 0, 1, 3, 3, 0, 1, 0, 2, 1, 0, 3)
def __call__(self, B, K, algorithm="encrypt"):
r"""
Apply S-DES encryption or decryption on the binary string ``B``
using the key ``K``. The flag ``algorithm`` controls what action is
to be performed on ``B``.
INPUT:
- ``B`` -- a binary string, where the number of bits is positive and
a multiple of 8.
- ``K`` -- a secret key; this must be a 10-bit binary string
- ``algorithm`` -- (default: ``"encrypt"``) a string; a flag to signify
whether encryption or decryption is to be applied to the binary
string ``B``. The encryption flag is ``"encrypt"`` and the decryption
flag is ``"decrypt"``.
OUTPUT:
- The ciphertext (respectively plaintext) corresponding to the
binary string ``B``.
EXAMPLES:
Encrypt a plaintext, decrypt the ciphertext, and compare the
result with the original plaintext::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: bin = BinaryStrings()
sage: P = bin.encoding("Encrypt this using DES!")
sage: K = sdes.random_key()
sage: K = sdes.list_to_string(K)
sage: C = sdes(P, K, algorithm="encrypt")
sage: plaintxt = sdes(C, K, algorithm="decrypt")
sage: plaintxt == P
True
TESTS:
The binary string ``B`` must be non-empty and the number of bits must
be a multiple of 8::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes("B", "K")
Traceback (most recent call last):
...
TypeError: input B must be a non-empty binary string with number of bits a multiple of 8
sage: bin = BinaryStrings()
sage: B = bin("101")
sage: sdes(B, "K")
Traceback (most recent call last):
...
ValueError: the number of bits in the binary string B must be positive and a multiple of 8
The secret key ``K`` must be a block of 10 bits::
sage: B = bin.encoding("abc")
sage: sdes(B, "K")
Traceback (most recent call last):
...
TypeError: secret key must be a 10-bit binary string
sage: K = bin("1010")
sage: sdes(B, K)
Traceback (most recent call last):
...
ValueError: secret key must be a 10-bit binary string
The value for ``algorithm`` must be either ``"encrypt"`` or
``"decrypt"``::
sage: B = bin.encoding("abc")
sage: K = sdes.list_to_string(sdes.random_key())
sage: sdes(B, K, algorithm="e")
Traceback (most recent call last):
...
ValueError: algorithm must be either 'encrypt' or 'decrypt'
sage: sdes(B, K, algorithm="d")
Traceback (most recent call last):
...
ValueError: algorithm must be either 'encrypt' or 'decrypt'
sage: sdes(B, K, algorithm="abc")
Traceback (most recent call last):
...
ValueError: algorithm must be either 'encrypt' or 'decrypt'
"""
from sage.monoids.string_monoid_element import StringMonoidElement
from sage.rings.finite_rings.integer_mod import Mod
Blength = 8
if not isinstance(B, StringMonoidElement):
raise TypeError, "input B must be a non-empty binary string with number of bits a multiple of 8"
if (len(B) == 0) or (Mod(len(B), Blength).lift() != 0):
raise ValueError, "the number of bits in the binary string B must be positive and a multiple of 8"
if not isinstance(K, StringMonoidElement):
raise TypeError, "secret key must be a 10-bit binary string"
if len(K) != self._key_size:
raise ValueError, "secret key must be a 10-bit binary string"
N = len(B) / Blength
S = ""
bin = BinaryStrings()
if algorithm == "encrypt":
for i in xrange(N):
block = B[i*Blength : (i+1)*Blength]
block = self.string_to_list(str(block))
key = self.string_to_list(str(K))
C = self.encrypt(block, key)
C = self.list_to_string(C)
S = "".join([S, str(C)])
return bin(S)
elif algorithm == "decrypt":
for i in xrange(N):
block = B[i*Blength : (i+1)*Blength]
block = self.string_to_list(str(block))
key = self.string_to_list(str(K))
P = self.decrypt(block, key)
P = self.list_to_string(P)
S = "".join([S, str(P)])
return bin(S)
else:
raise ValueError, "algorithm must be either 'encrypt' or 'decrypt'"
def __eq__(self, other):
r"""
Compare ``self`` with ``other``.
Simplified DES objects are the same if they have the same key size
and S-boxes.
EXAMPLES::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: s = SimplifiedDES()
sage: s == loads(dumps(s))
True
"""
return ( (self._key_size == other._key_size) and
(self._sbox0 == other._sbox0) and
(self._sbox1 == other._sbox1) )
def __repr__(self):
r"""
A string representation of this Simplified DES.
EXAMPLES::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: SimplifiedDES()
Simplified DES block cipher with 10-bit keys
"""
return "Simplified DES block cipher with 10-bit keys"
def block_length(self):
r"""
Return the block length of Schaefer's S-DES block cipher. A key in
Schaefer's S-DES is a block of 10 bits.
OUTPUT:
- The block (or key) length in number of bits.
EXAMPLES::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.block_length()
10
"""
return self._key_size
def decrypt(self, C, K):
r"""
Return an 8-bit plaintext corresponding to the ciphertext ``C``,
using S-DES decryption with key ``K``. The decryption process of
S-DES is as follows. Let `P` be the initial permutation function,
`P^{-1}` the corresponding inverse permutation, `\Pi_F` the
permutation/substitution function, and `\sigma` the switch function.
The ciphertext block ``C`` first goes through `P`, the output of
which goes through `\Pi_F` using the second subkey. Then we apply
the switch function to the output of the last function, and the
result is then fed into `\Pi_F` using the first subkey. Finally,
run the output through `P^{-1}` to get the plaintext.
INPUT:
- ``C`` -- an 8-bit ciphertext; a block of 8 bits
- ``K`` -- a 10-bit key; a block of 10 bits
OUTPUT:
The 8-bit plaintext corresponding to ``C``, obtained using the
key ``K``.
EXAMPLES:
Decrypt an 8-bit ciphertext block::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: C = [0, 1, 0, 1, 0, 1, 0, 1]
sage: K = [1, 0, 1, 0, 0, 0, 0, 0, 1, 0]
sage: sdes.decrypt(C, K)
[0, 0, 0, 1, 0, 1, 0, 1]
We can also work with strings of bits::
sage: C = "01010101"
sage: K = "1010000010"
sage: sdes.decrypt(sdes.string_to_list(C), sdes.string_to_list(K))
[0, 0, 0, 1, 0, 1, 0, 1]
TESTS:
The ciphertext must be a block of 8 bits::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.decrypt("C", "K")
Traceback (most recent call last):
...
TypeError: ciphertext must be a list of 8 bits
sage: sdes.decrypt([], "K")
Traceback (most recent call last):
...
ValueError: ciphertext must be a list of 8 bits
sage: sdes.decrypt([1, 2, 3, 4], "K")
Traceback (most recent call last):
...
ValueError: ciphertext must be a list of 8 bits
The key must be a block of 10 bits::
sage: sdes.decrypt([1, 0, 1, 0, 1, 1, 0, 1], "K")
Traceback (most recent call last):
...
TypeError: the key must be a list of 10 bits
sage: sdes.decrypt([1, 0, 1, 0, 1, 1, 0, 1], [])
Traceback (most recent call last):
...
TypeError: the key must be a list of 10 bits
sage: sdes.decrypt([1, 0, 1, 0, 1, 1, 0, 1], [1, 2, 3, 4, 5])
Traceback (most recent call last):
...
TypeError: the key must be a list of 10 bits
The value of each element of ``C`` or ``K`` must be either 0 or 1::
sage: C = [1, 2, 3, 4, 5, 6, 7, 8]
sage: K = [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
sage: sdes.decrypt(C, K)
Traceback (most recent call last):
...
TypeError: Argument x (= 2) is not a valid string.
sage: C = [0, 1, 0, 0, 1, 1, 1, 0]
sage: K = [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
sage: sdes.decrypt(C, K)
Traceback (most recent call last):
...
TypeError: Argument x (= 13) is not a valid string.
"""
if not isinstance(C, list):
raise TypeError, "ciphertext must be a list of 8 bits"
if len(C) != 8:
raise ValueError, "ciphertext must be a list of 8 bits"
if not isinstance(K, list):
raise TypeError, "the key must be a list of 10 bits"
if len(K) != 10:
raise TypeError, "the key must be a list of 10 bits"
P = self.initial_permutation(C, inverse=False)
P = self.permute_substitute(P, self.subkey(K, n=2))
P = self.switch(P)
P = self.permute_substitute(P, self.subkey(K, n=1))
P = self.initial_permutation(P, inverse=True)
return P
def encrypt(self, P, K):
r"""
Return an 8-bit ciphertext corresponding to the plaintext ``P``,
using S-DES encryption with key ``K``. The encryption process of
S-DES is as follows. Let `P` be the initial permutation function,
`P^{-1}` the corresponding inverse permutation, `\Pi_F` the
permutation/substitution function, and `\sigma` the switch function.
The plaintext block ``P`` first goes through `P`, the output of
which goes through `\Pi_F` using the first subkey. Then we apply
the switch function to the output of the last function, and the
result is then fed into `\Pi_F` using the second subkey. Finally,
run the output through `P^{-1}` to get the ciphertext.
INPUT:
- ``P`` -- an 8-bit plaintext; a block of 8 bits
- ``K`` -- a 10-bit key; a block of 10 bits
OUTPUT:
The 8-bit ciphertext corresponding to ``P``, obtained using the
key ``K``.
EXAMPLES:
Encrypt an 8-bit plaintext block::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: P = [0, 1, 0, 1, 0, 1, 0, 1]
sage: K = [1, 0, 1, 0, 0, 0, 0, 0, 1, 0]
sage: sdes.encrypt(P, K)
[1, 1, 0, 0, 0, 0, 0, 1]
We can also work with strings of bits::
sage: P = "01010101"
sage: K = "1010000010"
sage: sdes.encrypt(sdes.string_to_list(P), sdes.string_to_list(K))
[1, 1, 0, 0, 0, 0, 0, 1]
TESTS:
The plaintext must be a block of 8 bits::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.encrypt("P", "K")
Traceback (most recent call last):
...
TypeError: plaintext must be a list of 8 bits
sage: sdes.encrypt([], "K")
Traceback (most recent call last):
...
ValueError: plaintext must be a list of 8 bits
sage: sdes.encrypt([1, 2, 3, 4], "K")
Traceback (most recent call last):
...
ValueError: plaintext must be a list of 8 bits
The key must be a block of 10 bits::
sage: sdes.encrypt([1, 0, 1, 0, 1, 1, 0, 1], "K")
Traceback (most recent call last):
...
TypeError: the key must be a list of 10 bits
sage: sdes.encrypt([1, 0, 1, 0, 1, 1, 0, 1], [])
Traceback (most recent call last):
...
TypeError: the key must be a list of 10 bits
sage: sdes.encrypt([1, 0, 1, 0, 1, 1, 0, 1], [1, 2, 3, 4, 5])
Traceback (most recent call last):
...
TypeError: the key must be a list of 10 bits
The value of each element of ``P`` or ``K`` must be either 0 or 1::
sage: P = [1, 2, 3, 4, 5, 6, 7, 8]
sage: K = [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
sage: sdes.encrypt(P, K)
Traceback (most recent call last):
...
TypeError: Argument x (= 2) is not a valid string.
sage: P = [0, 1, 0, 0, 1, 1, 1, 0]
sage: K = [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
sage: sdes.encrypt(P, K)
Traceback (most recent call last):
...
TypeError: Argument x (= 13) is not a valid string.
"""
if not isinstance(P, list):
raise TypeError, "plaintext must be a list of 8 bits"
if len(P) != 8:
raise ValueError, "plaintext must be a list of 8 bits"
if not isinstance(K, list):
raise TypeError, "the key must be a list of 10 bits"
if len(K) != 10:
raise TypeError, "the key must be a list of 10 bits"
C = self.initial_permutation(P, inverse=False)
C = self.permute_substitute(C, self.subkey(K, n=1))
C = self.switch(C)
C = self.permute_substitute(C, self.subkey(K, n=2))
C = self.initial_permutation(C, inverse=True)
return C
def initial_permutation(self, B, inverse=False):
r"""
Return the initial permutation of ``B``. Denote the initial
permutation function by `P` and let `(b_0, b_1, b_2, \dots, b_7)`
be a vector of 8 bits, where each `b_i \in \{ 0, 1 \}`. Then
.. MATH::
P(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7)
= (b_1, b_5, b_2, b_0, b_3, b_7, b_4, b_6)
The inverse permutation is `P^{-1}`:
.. MATH::
P^{-1}(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7)
= (b_3, b_0, b_2, b_4, b_6, b_1, b_7, b_5)
INPUT:
- ``B`` -- list; a block of 8 bits
- ``inverse`` -- (default: ``False``) if ``True`` then use the
inverse permutation `P^{-1}`; if ``False`` then use the initial
permutation `P`
OUTPUT:
The initial permutation of ``B`` if ``inverse=False``, or the
inverse permutation of ``B`` if ``inverse=True``.
EXAMPLES:
The initial permutation of a list of 8 bits::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: B = [1, 0, 1, 1, 0, 1, 0, 0]
sage: P = sdes.initial_permutation(B); P
[0, 1, 1, 1, 1, 0, 0, 0]
Recovering the original list of 8 bits from the permutation::
sage: Pinv = sdes.initial_permutation(P, inverse=True)
sage: Pinv; B
[1, 0, 1, 1, 0, 1, 0, 0]
[1, 0, 1, 1, 0, 1, 0, 0]
We can also work with a string of bits::
sage: S = "10110100"
sage: L = sdes.string_to_list(S)
sage: P = sdes.initial_permutation(L); P
[0, 1, 1, 1, 1, 0, 0, 0]
sage: sdes.initial_permutation(sdes.string_to_list("01111000"), inverse=True)
[1, 0, 1, 1, 0, 1, 0, 0]
TESTS:
The input block must be a list::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.initial_permutation("B")
Traceback (most recent call last):
...
TypeError: input block must be a list of 8 bits
sage: sdes.initial_permutation(())
Traceback (most recent call last):
...
TypeError: input block must be a list of 8 bits
The input block must be a list of 8 bits::
sage: sdes.initial_permutation([])
Traceback (most recent call last):
...
ValueError: input block must be a list of 8 bits
sage: sdes.initial_permutation([1, 2, 3, 4, 5, 6, 7, 8, 9])
Traceback (most recent call last):
...
ValueError: input block must be a list of 8 bits
The value of each element of the list must be either 0 or 1::
sage: sdes.initial_permutation([1, 2, 3, 4, 5, 6, 7, 8])
Traceback (most recent call last):
...
TypeError: Argument x (= 2) is not a valid string.
"""
if not isinstance(B, list):
raise TypeError, "input block must be a list of 8 bits"
if len(B) != 8:
raise ValueError, "input block must be a list of 8 bits"
bin = BinaryStrings()
if not inverse:
return [ bin(str(B[1])), bin(str(B[5])),
bin(str(B[2])), bin(str(B[0])),
bin(str(B[3])), bin(str(B[7])),
bin(str(B[4])), bin(str(B[6])) ]
if inverse:
return [ bin(str(B[3])), bin(str(B[0])),
bin(str(B[2])), bin(str(B[4])),
bin(str(B[6])), bin(str(B[1])),
bin(str(B[7])), bin(str(B[5])) ]
def left_shift(self, B, n=1):
r"""
Return a circular left shift of ``B`` by ``n`` positions. Let
`B = (b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8, b_9)` be a vector
of 10 bits. Then the left shift operation `L_n` is performed on the
first 5 bits and the last 5 bits of `B` separately. That is, if the
number of shift positions is ``n=1``, then `L_1` is defined as
.. MATH::
L_1(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8, b_9)
= (b_1, b_2, b_3, b_4, b_0, b_6, b_7, b_8, b_9, b_5)
If the number of shift positions is ``n=2``, then `L_2` is given by
.. MATH::
L_2(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8, b_9)
= (b_2, b_3, b_4, b_0, b_1, b_7, b_8, b_9, b_5, b_6)
INPUT:
- ``B`` -- a list of 10 bits
- ``n`` -- (default: 1) if ``n=1`` then perform left shift by 1
position; if ``n=2`` then perform left shift by 2 positions. The
valid values for ``n`` are 1 and 2, since only up to 2 positions
are defined for this circular left shift operation.
OUTPUT:
The circular left shift of each half of ``B``.
EXAMPLES:
Circular left shift by 1 position of a 10-bit string::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: B = [1, 0, 0, 0, 0, 0, 1, 1, 0, 0]
sage: sdes.left_shift(B)
[0, 0, 0, 0, 1, 1, 1, 0, 0, 0]
sage: sdes.left_shift([1, 0, 1, 0, 0, 0, 0, 0, 1, 0])
[0, 1, 0, 0, 1, 0, 0, 1, 0, 0]
Circular left shift by 2 positions of a 10-bit string::
sage: B = [0, 0, 0, 0, 1, 1, 1, 0, 0, 0]
sage: sdes.left_shift(B, n=2)
[0, 0, 1, 0, 0, 0, 0, 0, 1, 1]
Here we work with a string of bits::
sage: S = "1000001100"
sage: L = sdes.string_to_list(S)
sage: sdes.left_shift(L)
[0, 0, 0, 0, 1, 1, 1, 0, 0, 0]
sage: sdes.left_shift(sdes.string_to_list("1010000010"), n=2)
[1, 0, 0, 1, 0, 0, 1, 0, 0, 0]
TESTS:
The input block must be a list::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.left_shift("B")
Traceback (most recent call last):
...
TypeError: input block must be a list of 10 bits
sage: sdes.left_shift(())
Traceback (most recent call last):
...
TypeError: input block must be a list of 10 bits
The input block must be a list of 10 bits::
sage: sdes.left_shift([])
Traceback (most recent call last):
...
ValueError: input block must be a list of 10 bits
sage: sdes.left_shift([1, 2, 3, 4, 5])
Traceback (most recent call last):
...
ValueError: input block must be a list of 10 bits
The value of each element of the list must be either 0 or 1::
sage: sdes.left_shift([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
Traceback (most recent call last):
...
TypeError: Argument x (= 2) is not a valid string.
The number of shift positions must be either 1 or 2::
sage: B = [0, 0, 0, 0, 1, 1, 1, 0, 0, 0]
sage: sdes.left_shift(B, n=-1)
Traceback (most recent call last):
...
ValueError: input n must be either 1 or 2
sage: sdes.left_shift(B, n=3)
Traceback (most recent call last):
...
ValueError: input n must be either 1 or 2
"""
if not isinstance(B, list):
raise TypeError, "input block must be a list of 10 bits"
if len(B) != 10:
raise ValueError, "input block must be a list of 10 bits"
bin = BinaryStrings()
if n == 1:
return [ bin(str(B[1])), bin(str(B[2])),
bin(str(B[3])), bin(str(B[4])),
bin(str(B[0])), bin(str(B[6])),
bin(str(B[7])), bin(str(B[8])),
bin(str(B[9])), bin(str(B[5])) ]
elif n == 2:
return [ bin(str(B[2])), bin(str(B[3])),
bin(str(B[4])), bin(str(B[0])),
bin(str(B[1])), bin(str(B[7])),
bin(str(B[8])), bin(str(B[9])),
bin(str(B[5])), bin(str(B[6])) ]
else:
raise ValueError, "input n must be either 1 or 2"
def list_to_string(self, B):
r"""
Return a binary string representation of the list ``B``.
INPUT:
- ``B`` -- a non-empty list of bits
OUTPUT:
The binary string representation of ``B``.
EXAMPLES:
A binary string representation of a list of bits::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: L = [0, 0, 0, 0, 1, 1, 0, 1, 0, 0]
sage: sdes.list_to_string(L)
0000110100
TESTS:
Input ``B`` must be a non-empty list::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.list_to_string("L")
Traceback (most recent call last):
...
TypeError: input B must be a non-empty list of bits
sage: sdes.list_to_string([])
Traceback (most recent call last):
...
ValueError: input B must be a non-empty list of bits
Input must be a non-empty list of bits::
sage: sdes.list_to_string([0, 1, 2])
Traceback (most recent call last):
...
IndexError: tuple index out of range
"""
if not isinstance(B, list):
raise TypeError, "input B must be a non-empty list of bits"
if len(B) == 0:
raise ValueError, "input B must be a non-empty list of bits"
from sage.rings.integer import Integer
bin = BinaryStrings()
return bin([Integer(str(b)) for b in B])
def permutation4(self, B):
r"""
Return a permutation of a 4-bit string. This permutation is called
`P_4` and is specified as follows. Let
`(b_0, b_1, b_2, b_3)` be a vector of 4 bits where each
`b_i \in \{ 0, 1 \}`. Then `P_4` is defined by
.. MATH::
P_4(b_0, b_1, b_2, b_3) = (b_1, b_3, b_2, b_0)
INPUT:
- ``B`` -- a block of 4-bit string
OUTPUT:
A permutation of ``B``.
EXAMPLES:
Permute a 4-bit string::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: B = [1, 1, 0, 0]
sage: sdes.permutation4(B)
[1, 0, 0, 1]
sage: sdes.permutation4([0, 1, 0, 1])
[1, 1, 0, 0]
We can also work with a string of bits::
sage: S = "1100"
sage: L = sdes.string_to_list(S)
sage: sdes.permutation4(L)
[1, 0, 0, 1]
sage: sdes.permutation4(sdes.string_to_list("0101"))
[1, 1, 0, 0]
TESTS:
The input block must be a list::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.permutation4("B")
Traceback (most recent call last):
...
TypeError: input block must be a list of 4 bits
sage: sdes.permutation4(())
Traceback (most recent call last):
...
TypeError: input block must be a list of 4 bits
The input block must be a list of 4 bits::
sage: sdes.permutation4([])
Traceback (most recent call last):
...
ValueError: input block must be a list of 4 bits
sage: sdes.permutation4([1, 2, 3, 4, 5])
Traceback (most recent call last):
...
ValueError: input block must be a list of 4 bits
The value of each element of the list must be either 0 or 1::
sage: sdes.permutation4([1, 2, 3, 4])
Traceback (most recent call last):
...
TypeError: Argument x (= 2) is not a valid string.
"""
if not isinstance(B, list):
raise TypeError, "input block must be a list of 4 bits"
if len(B) != 4:
raise ValueError, "input block must be a list of 4 bits"
bin = BinaryStrings()
return [ bin(str(B[1])), bin(str(B[3])),
bin(str(B[2])), bin(str(B[0])) ]
def permutation8(self, B):
r"""
Return a permutation of an 8-bit string. This permutation is called
`P_8` and is specified as follows. Let
`(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8, b_9)` be a vector of
10 bits where each `b_i \in \{ 0, 1 \}`. Then `P_8` picks out 8 of
those 10 bits and permutes those 8 bits:
.. MATH::
P_8(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8, b_9)
=
(b_5, b_2, b_6, b_3, b_7, b_4, b_9, b_8)
INPUT:
- ``B`` -- a block of 10-bit string
OUTPUT:
Pick out 8 of the 10 bits of ``B`` and permute those 8 bits.
EXAMPLES:
Permute a 10-bit string::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: B = [1, 1, 0, 0, 1, 0, 0, 1, 0, 1]
sage: sdes.permutation8(B)
[0, 0, 0, 0, 1, 1, 1, 0]
sage: sdes.permutation8([0, 1, 1, 0, 1, 0, 0, 1, 0, 1])
[0, 1, 0, 0, 1, 1, 1, 0]
sage: sdes.permutation8([0, 0, 0, 0, 1, 1, 1, 0, 0, 0])
[1, 0, 1, 0, 0, 1, 0, 0]
We can also work with a string of bits::
sage: S = "1100100101"
sage: L = sdes.string_to_list(S)
sage: sdes.permutation8(L)
[0, 0, 0, 0, 1, 1, 1, 0]
sage: sdes.permutation8(sdes.string_to_list("0110100101"))
[0, 1, 0, 0, 1, 1, 1, 0]
TESTS:
The input block must be a list::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.permutation8("B")
Traceback (most recent call last):
...
TypeError: input block must be a list of 10 bits
sage: sdes.permutation8(())
Traceback (most recent call last):
...
TypeError: input block must be a list of 10 bits
The input block must be a list of 10 bits::
sage: sdes.permutation8([])
Traceback (most recent call last):
...
ValueError: input block must be a list of 10 bits
sage: sdes.permutation8([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])
Traceback (most recent call last):
...
ValueError: input block must be a list of 10 bits
The value of each element of the list must be either 0 or 1::
sage: sdes.permutation8([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
Traceback (most recent call last):
...
TypeError: Argument x (= 6) is not a valid string.
"""
if not isinstance(B, list):
raise TypeError, "input block must be a list of 10 bits"
if len(B) != 10:
raise ValueError, "input block must be a list of 10 bits"
bin = BinaryStrings()
return [ bin(str(B[5])), bin(str(B[2])),
bin(str(B[6])), bin(str(B[3])),
bin(str(B[7])), bin(str(B[4])),
bin(str(B[9])), bin(str(B[8])) ]
def permutation10(self, B):
r"""
Return a permutation of a 10-bit string. This permutation is called
`P_{10}` and is specified as follows. Let
`(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8, b_9)` be a vector of
10 bits where each `b_i \in \{ 0, 1 \}`. Then `P_{10}` is given by
.. MATH::
P_{10}(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7, b_8, b_9)
=
(b_2, b_4, b_1, b_6, b_3, b_9, b_0, b_8, b_7, b_5)
INPUT:
- ``B`` -- a block of 10-bit string
OUTPUT:
A permutation of ``B``.
EXAMPLES:
Permute a 10-bit string::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: B = [1, 1, 0, 0, 1, 0, 0, 1, 0, 1]
sage: sdes.permutation10(B)
[0, 1, 1, 0, 0, 1, 1, 0, 1, 0]
sage: sdes.permutation10([0, 1, 1, 0, 1, 0, 0, 1, 0, 1])
[1, 1, 1, 0, 0, 1, 0, 0, 1, 0]
sage: sdes.permutation10([1, 0, 1, 0, 0, 0, 0, 0, 1, 0])
[1, 0, 0, 0, 0, 0, 1, 1, 0, 0]
Here we work with a string of bits::
sage: S = "1100100101"
sage: L = sdes.string_to_list(S)
sage: sdes.permutation10(L)
[0, 1, 1, 0, 0, 1, 1, 0, 1, 0]
sage: sdes.permutation10(sdes.string_to_list("0110100101"))
[1, 1, 1, 0, 0, 1, 0, 0, 1, 0]
TESTS:
The input block must be a list::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.permutation10("B")
Traceback (most recent call last):
...
TypeError: input block must be a list of 10 bits
sage: sdes.permutation10(())
Traceback (most recent call last):
...
TypeError: input block must be a list of 10 bits
The input block must be a list of 10 bits::
sage: sdes.permutation10([])
Traceback (most recent call last):
...
ValueError: input block must be a list of 10 bits
sage: sdes.permutation10([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11])
Traceback (most recent call last):
...
ValueError: input block must be a list of 10 bits
The value of each element of the list must be either 0 or 1::
sage: sdes.permutation10([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
Traceback (most recent call last):
...
TypeError: Argument x (= 3) is not a valid string.
"""
if not isinstance(B, list):
raise TypeError, "input block must be a list of 10 bits"
if len(B) != 10:
raise ValueError, "input block must be a list of 10 bits"
bin = BinaryStrings()
return [ bin(str(B[2])), bin(str(B[4])),
bin(str(B[1])), bin(str(B[6])),
bin(str(B[3])), bin(str(B[9])),
bin(str(B[0])), bin(str(B[8])),
bin(str(B[7])), bin(str(B[5])) ]
def permute_substitute(self, B, key):
r"""
Apply the function `\Pi_F` on the block ``B`` using subkey ``key``.
Let `(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7)`
be a vector of 8 bits where each `b_i \in \{ 0, 1 \}`, let `L` and
`R` be the leftmost 4 bits and rightmost 4 bits of ``B``
respectively, and let `F` be a function mapping 4-bit strings to
4-bit strings. Then
.. MATH::
\Pi_F(L, R) = (L \oplus F(R, S), R)
where `S` is a subkey and `\oplus` denotes the bit-wise
exclusive-OR function.
The function `F` can be described as follows. Its 4-bit input block
`(n_0, n_1, n_2, n_3)` is first expanded into an 8-bit block
to become `(n_3, n_0, n_1, n_2, n_1, n_2, n_3, n_0)`. This is
usually represented as follows
.. MATH::
\begin{tabular}{c|cc|c}
$n_3$ & $n_0$ & $n_1$ & $n_2$ \\
$n_1$ & $n_2$ & $n_3$ & $n_0$
\end{tabular}
Let `K = (k_0, k_1, k_2, k_3, k_4, k_5, k_6, k_7)` be an 8-bit
subkey. Then `K` is added to the above expanded input block using
exclusive-OR to produce
.. MATH::
\begin{tabular}{c|cc|c}
$n_3 + k_0$ & $n_0 + k_1$ & $n_1 + k_2$ & $n_2 + k_3$ \\
$n_1 + k_4$ & $n_2 + k_5$ & $n_3 + k_6$ & $n_0 + k_7$
\end{tabular}
=
\begin{tabular}{c|cc|c}
$p_{0,0}$ & $p_{0,1}$ & $p_{0,2}$ & $p_{0,3}$ \\
$p_{1,0}$ & $p_{1,1}$ & $p_{1,2}$ & $p_{1,3}$
\end{tabular}
Now read the first row as the 4-bit string
`p_{0,0} p_{0,3} p_{0,1} p_{0,2}` and input this 4-bit string through
S-box `S_0` to get a 2-bit output.
.. MATH::
S_0
=
\begin{tabular}{cc|cc} \hline
Input & Output & Input & Output \\\hline
0000 & 01 & 1000 & 00 \\
0001 & 00 & 1001 & 10 \\
0010 & 11 & 1010 & 01 \\
0011 & 10 & 1011 & 11 \\
0100 & 11 & 1100 & 11 \\
0101 & 10 & 1101 & 01 \\
0110 & 01 & 1110 & 11 \\
0111 & 00 & 1111 & 10 \\\hline
\end{tabular}
Next read the second row as the 4-bit string
`p_{1,0} p_{1,3} p_{1,1} p_{1,2}` and input this 4-bit string through
S-box `S_1` to get another 2-bit output.
.. MATH::
S_1
=
\begin{tabular}{cc|cc} \hline
Input & Output & Input & Output \\\hline
0000 & 00 & 1000 & 11 \\
0001 & 01 & 1001 & 00 \\
0010 & 10 & 1010 & 01 \\
0011 & 11 & 1011 & 00 \\
0100 & 10 & 1100 & 10 \\
0101 & 00 & 1101 & 01 \\
0110 & 01 & 1110 & 00 \\
0111 & 11 & 1111 & 11 \\\hline
\end{tabular}
Denote the 4 bits produced by `S_0` and `S_1` as `b_0 b_1 b_2 b_3`.
This 4-bit string undergoes another permutation called `P_4` as
follows:
.. MATH::
P_4(b_0, b_1, b_2, b_3) = (b_1, b_3, b_2, b_0)
The output of `P_4` is the output of the function `F`.
INPUT:
- ``B`` -- a list of 8 bits
- ``key`` -- an 8-bit subkey
OUTPUT:
The result of applying the function `\Pi_F` to ``B``.
EXAMPLES:
Applying the function `\Pi_F` to an 8-bit block and an 8-bit subkey::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: B = [1, 0, 1, 1, 1, 1, 0, 1]
sage: K = [1, 1, 0, 1, 0, 1, 0, 1]
sage: sdes.permute_substitute(B, K)
[1, 0, 1, 0, 1, 1, 0, 1]
We can also work with strings of bits::
sage: B = "10111101"
sage: K = "11010101"
sage: B = sdes.string_to_list(B); K = sdes.string_to_list(K)
sage: sdes.permute_substitute(B, K)
[1, 0, 1, 0, 1, 1, 0, 1]
TESTS:
The input ``B`` must be a block of 8 bits::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.permute_substitute("B", "K")
Traceback (most recent call last):
...
TypeError: input B must be an 8-bit string
sage: sdes.permute_substitute([], "K")
Traceback (most recent call last):
...
ValueError: input B must be an 8-bit string
The input ``key`` must be an 8-bit subkey::
sage: sdes.permute_substitute([0, 1, 0, 0, 1, 1, 1, 0], "K")
Traceback (most recent call last):
...
TypeError: input key must be an 8-bit subkey
sage: sdes.permute_substitute([0, 1, 0, 0, 1, 1, 1, 0], [])
Traceback (most recent call last):
...
ValueError: input key must be an 8-bit subkey
The value of each element of ``B`` or ``key`` must be either 0 or 1::
sage: B = [1, 2, 3, 4, 5, 6, 7, 8]
sage: K = [0, 1, 2, 3, 4, 5, 6, 7]
sage: sdes.permute_substitute(B, K)
Traceback (most recent call last):
...
TypeError: Argument x (= 2) is not a valid string.
sage: B = [0, 1, 0, 0, 1, 1, 1, 0]
sage: K = [1, 2, 3, 4, 5, 6, 7, 8]
sage: sdes.permute_substitute(B, K)
Traceback (most recent call last):
...
TypeError: Argument x (= 2) is not a valid string.
"""
if not isinstance(B, list):
raise TypeError, "input B must be an 8-bit string"
if len(B) != 8:
raise ValueError, "input B must be an 8-bit string"
if not isinstance(key, list):
raise TypeError, "input key must be an 8-bit subkey"
if len(key) != 8:
raise ValueError, "input key must be an 8-bit subkey"
from sage.rings.finite_rings.constructor import FiniteField
GF = FiniteField(2, "x")
bin = BinaryStrings()
bin_to_GF2 = {bin("0"): GF(0), bin("1"): GF(1)}
L = [ bin_to_GF2[bin(str(B[i]))] for i in xrange(4) ]
R = [ bin_to_GF2[bin(str(B[i]))] for i in xrange(4, len(B)) ]
K = [ bin_to_GF2[bin(str(key[i]))] for i in xrange(len(key)) ]
RX = [ R[3], R[0], R[1], R[2], R[1], R[2], R[3], R[0] ]
P = [ RX[i] + K[i] for i in xrange(len(K)) ]
left = self._sbox0([ P[0], P[3], P[1], P[2] ])
right = self._sbox1([ P[4], P[7], P[5], P[6] ])
F = self.permutation4(left + right)
F = [ bin_to_GF2[F[i]] for i in xrange(len(F)) ]
L = [ L[i] + F[i] for i in xrange(len(F)) ]
return L + R
def random_key(self):
r"""
Return a random 10-bit key.
EXAMPLES:
The size of each key is the same as the block size::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: key = sdes.random_key()
sage: len(key) == sdes.block_length()
True
"""
from sage.misc.prandom import randint
bin = BinaryStrings()
return [bin(str(randint(0, 1))) for i in xrange(self._key_size)]
def sbox(self):
r"""
Return the S-boxes of simplified DES.
EXAMPLES::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sbox = sdes.sbox()
sage: sbox[0]; sbox[1]
(1, 0, 3, 2, 3, 2, 1, 0, 0, 2, 1, 3, 3, 1, 3, 2)
(0, 1, 2, 3, 2, 0, 1, 3, 3, 0, 1, 0, 2, 1, 0, 3)
"""
return [self._sbox0, self._sbox1]
def string_to_list(self, S):
r"""
Return a list representation of the binary string ``S``.
INPUT:
- ``S`` -- a string of bits
OUTPUT:
A list representation of the string ``S``.
EXAMPLES:
A list representation of a string of bits::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: S = "0101010110"
sage: sdes.string_to_list(S)
[0, 1, 0, 1, 0, 1, 0, 1, 1, 0]
TESTS:
Input must be a non-empty string::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.string_to_list("")
Traceback (most recent call last):
...
ValueError: input S must be a non-empty string of bits
sage: sdes.string_to_list(1)
Traceback (most recent call last):
...
TypeError: input S must be a non-empty string of bits
Input must be a non-empty string of bits::
sage: sdes.string_to_list("0123")
Traceback (most recent call last):
...
TypeError: Argument x (= 2) is not a valid string.
"""
if not isinstance(S, str):
raise TypeError, "input S must be a non-empty string of bits"
if len(S) == 0:
raise ValueError, "input S must be a non-empty string of bits"
bin = BinaryStrings()
return [bin(s) for s in S]
def subkey(self, K, n=1):
r"""
Return the ``n``-th subkey based on the key ``K``.
INPUT:
- ``K`` -- a 10-bit secret key of this Simplified DES
- ``n`` -- (default: 1) if ``n=1`` then return the first subkey based
on ``K``; if ``n=2`` then return the second subkey. The valid
values for ``n`` are 1 and 2, since only two subkeys are defined
for each secret key in Schaefer's S-DES.
OUTPUT:
The ``n``-th subkey based on the secret key ``K``.
EXAMPLES:
Obtain the first subkey from a secret key::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: key = [1, 0, 1, 0, 0, 0, 0, 0, 1, 0]
sage: sdes.subkey(key, n=1)
[1, 0, 1, 0, 0, 1, 0, 0]
Obtain the second subkey from a secret key::
sage: key = [1, 0, 1, 0, 0, 0, 0, 0, 1, 0]
sage: sdes.subkey(key, n=2)
[0, 1, 0, 0, 0, 0, 1, 1]
We can also work with strings of bits::
sage: K = "1010010010"
sage: L = sdes.string_to_list(K)
sage: sdes.subkey(L, n=1)
[1, 0, 1, 0, 0, 1, 0, 1]
sage: sdes.subkey(sdes.string_to_list("0010010011"), n=2)
[0, 1, 1, 0, 1, 0, 1, 0]
TESTS:
Input ``K`` must be a 10-bit key::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.subkey("K")
Traceback (most recent call last):
...
TypeError: input K must be a 10-bit key
sage: sdes.subkey([])
Traceback (most recent call last):
...
ValueError: input K must be a 10-bit key
There are only two subkeys::
sage: key = [1, 0, 1, 0, 0, 0, 0, 0, 1, 0]
sage: sdes.subkey(key, n=0)
Traceback (most recent call last):
...
ValueError: input n must be either 1 or 2
sage: sdes.subkey(key, n=3)
Traceback (most recent call last):
...
ValueError: input n must be either 1 or 2
"""
if not isinstance(K, list):
raise TypeError, "input K must be a 10-bit key"
if len(K) != self._key_size:
raise ValueError, "input K must be a 10-bit key"
if n == 1:
key1 = self.permutation10(K)
key1 = self.left_shift(key1, n=1)
return self.permutation8(key1)
elif n == 2:
key2 = self.permutation10(K)
key2 = self.left_shift(key2, n=1)
key2 = self.left_shift(key2, n=2)
return self.permutation8(key2)
else:
raise ValueError, "input n must be either 1 or 2"
def switch(self, B):
r"""
Interchange the first 4 bits with the last 4 bits in the list ``B``
of 8 bits. Let `(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7)`
be a vector of 8 bits, where each `b_i \in \{ 0, 1 \}`. Then the
switch function `\sigma` is given by
.. MATH::
\sigma(b_0, b_1, b_2, b_3, b_4, b_5, b_6, b_7)
= (b_4, b_5, b_6, b_7, b_0, b_1, b_2, b_3)
INPUT:
- ``B`` -- list; a block of 8 bits
OUTPUT:
A block of the same dimension, but in which the first 4 bits from
``B`` has been switched for the last 4 bits in ``B``.
EXAMPLES:
Interchange the first 4 bits with the last 4 bits::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: B = [1, 1, 1, 0, 1, 0, 0, 0]
sage: sdes.switch(B)
[1, 0, 0, 0, 1, 1, 1, 0]
sage: sdes.switch([1, 1, 1, 1, 0, 0, 0, 0])
[0, 0, 0, 0, 1, 1, 1, 1]
We can also work with a string of bits::
sage: S = "11101000"
sage: L = sdes.string_to_list(S)
sage: sdes.switch(L)
[1, 0, 0, 0, 1, 1, 1, 0]
sage: sdes.switch(sdes.string_to_list("11110000"))
[0, 0, 0, 0, 1, 1, 1, 1]
TESTS:
The input block must be a list::
sage: from sage.crypto.block_cipher.sdes import SimplifiedDES
sage: sdes = SimplifiedDES()
sage: sdes.switch("B")
Traceback (most recent call last):
...
TypeError: input block must be a list of 8 bits
sage: sdes.switch(())
Traceback (most recent call last):
...
TypeError: input block must be a list of 8 bits
The input block must be a list of 8 bits::
sage: sdes.switch([])
Traceback (most recent call last):
...
ValueError: input block must be a list of 8 bits
sage: sdes.switch([1, 2, 3, 4, 5, 6, 7, 8, 9])
Traceback (most recent call last):
...
ValueError: input block must be a list of 8 bits
The value of each element of the list must be either 0 or 1::
sage: sdes.switch([1, 2, 3, 4, 5, 6, 7, 8])
Traceback (most recent call last):
...
TypeError: Argument x (= 5) is not a valid string.
"""
if not isinstance(B, list):
raise TypeError, "input block must be a list of 8 bits"
if len(B) != 8:
raise ValueError, "input block must be a list of 8 bits"
bin = BinaryStrings()
return [ bin(str(B[4])), bin(str(B[5])),
bin(str(B[6])), bin(str(B[7])),
bin(str(B[0])), bin(str(B[1])),
bin(str(B[2])), bin(str(B[3])) ]
