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

class sage.crypto.block_cipher.sdes.SimplifiedDES#

Bases: SageObject

This class implements the Simplified Data Encryption Standard (S-DES) described in [Sch1996]. 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 Mini-AES [Pha2002].

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 range(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

block_length()#

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

decrypt(C, K)#

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, $Π_{F}$ the permutation/substitution function, and $σ$ the switch function. The ciphertext block C first goes through $P$ , the output of which goes through $Π_{F}$ using the second subkey. Then we apply the switch function to the output of the last function, and the result is then fed into $Π_{F}$ 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]

encrypt(P, K)#

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, $Π_{F}$ the permutation/substitution function, and $σ$ the switch function. The plaintext block P first goes through $P$ , the output of which goes through $Π_{F}$ using the first subkey. Then we apply the switch function to the output of the last function, and the result is then fed into $Π_{F}$ 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]

initial_permutation(B, inverse=False)#

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

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 $P^{- 1}$ :

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]

left_shift(B, n=1)#

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

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

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]

list_to_string(B)#

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

permutation10(B)#

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

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]

permutation4(B)#

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

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]

permutation8(B)#

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:

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]

permute_substitute(B, key)#

Apply the function $Π_{F}$ 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

Π_{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

Unknown environment '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

Unknown environment '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.

Unknown environment '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.

Unknown environment '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:

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 $Π_{F}$ to B.

EXAMPLES:

Applying the function $Π_{F}$ 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]

random_key()#

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

sbox()#

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)

string_to_list(S)#

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]

subkey(K, n=1)#

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]

switch(B)#

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 $σ$ is given by

σ (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]