Mini-AES

A simplified variant of the Advanced Encryption Standard (AES). Note that Mini-AES is for educational purposes only. It is a small-scale version of the AES designed to help beginners understand the basic structure of AES.

AUTHORS:

• Minh Van Nguyen (2009-05): initial version

class sage.crypto.block_cipher.miniaes.MiniAES

Bases: SageObject

This class implements the Mini Advanced Encryption Standard (Mini-AES) described in [Pha2002]. Note that Phan’s Mini-AES is for educational purposes only and is not secure for practical purposes. Mini-AES is a version of the AES with all parameters significantly reduced, but at the same time preserving the structure of AES. The goal of Mini-AES is to allow a beginner to understand the structure of AES, thus laying a foundation for a thorough study of AES. Its goal is as a teaching tool and is different from the SR small scale variants of the AES. SR defines a family of parameterizable variants of the AES suitable as a framework for comparing different cryptanalytic techniques that can be brought to bear on the AES.

EXAMPLES:

Encrypt a plaintext:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: P = MS([K("x^3 + x"), K("x^2 + 1"), K("x^2 + x"), K("x^3 + x^2")]); P

[  x^3 + x   x^2 + 1]
[  x^2 + x x^3 + x^2]
sage: key = MS([K("x^3 + x^2"), K("x^3 + x"), K("x^3 + x^2 + x"), K("x^2 + x + 1")]); key

[    x^3 + x^2       x^3 + x]
[x^3 + x^2 + x   x^2 + x + 1]
sage: C = maes.encrypt(P, key); C
[    x^3 + x^2 + 1 x^3 + x^2 + x + 1]
[                x     x^3 + x^2 + x]

Decrypt the result:

sage: plaintxt = maes.decrypt(C, key)
sage: plaintxt; P

[  x^3 + x   x^2 + 1]
[  x^2 + x x^3 + x^2]

[  x^3 + x   x^2 + 1]
[  x^2 + x x^3 + x^2]
sage: plaintxt == P
True

We can also work directly with binary strings:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: bin = BinaryStrings()
sage: key = bin.encoding("KE"); key
0100101101000101
sage: P = bin.encoding("Encrypt this secret message!"); P
01000101011011100110001101110010011110010111000001110100001000000111010001101000011010010111001100100000011100110110010101100011011100100110010101110100001000000110110101100101011100110111001101100001011001110110010100100001
sage: C = maes(P, key, algorithm="encrypt"); C
11100000101000010110001101101001110110010010111011010001100111100000101000101111100110010010100001110101011100111001000010101000001111000101010011010001100111100111001100000001101100110110101001001000011100000101010110110101
sage: plaintxt = maes(C, key, algorithm="decrypt")
sage: plaintxt == P
True

Now we work with integers \(n\) such that \(0 \leq n \leq 15\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: P = [n for n in range(16)]; P
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
sage: key = [2, 3, 11, 0]; key
[2, 3, 11, 0]
sage: P = maes.integer_to_binary(P); P
0000000100100011010001010110011110001001101010111100110111101111
sage: key = maes.integer_to_binary(key); key
0010001110110000
sage: C = maes(P, key, algorithm="encrypt"); C
0011101000101110010011000101010100011101010000111000100100011010
sage: plaintxt = maes(C, key, algorithm="decrypt")
sage: plaintxt == P
True

Generate some random plaintext and a random secret key. Encrypt the plaintext using that secret key and decrypt the result. Then compare the decrypted plaintext with the original plaintext:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: MS = MatrixSpace(FiniteField(16, "x"), 2, 2)
sage: P = MS.random_element()
sage: key = maes.random_key()
sage: C = maes.encrypt(P, key)
sage: plaintxt = maes.decrypt(C, key)
sage: plaintxt == P
True

GF_to_binary(G)

Return the binary representation of G. If G is an element of the finite field \(\GF{2^4}\), then obtain the binary representation of G. If G is a list of elements belonging to \(\GF{2^4}\), obtain the 4-bit representation of each element of the list, then concatenate the resulting 4-bit strings into a binary string. If G is a matrix with entries over \(\GF{2^4}\), convert each matrix entry to its 4-bit representation, then concatenate the 4-bit strings. The concatenation is performed starting from the top-left corner of the matrix, working across left to right, top to bottom. Each element of \(\GF{2^4}\) can be associated with a unique 4-bit string according to the following table:

\[\begin{split}\begin{tabular}{ll|ll} \hline 4-bit string & $\GF{2^4}$ & 4-bit string & $\GF{2^4}$ \\\hline 0000 & $0$ & 1000 & $x^3$ \\ 0001 & $1$ & 1001 & $x^3 + 1$ \\ 0010 & $x$ & 1010 & $x^3 + x$ \\ 0011 & $x + 1$ & 1011 & $x^3 + x + 1$ \\ 0100 & $x^2$ & 1100 & $x^3 + x^2$ \\ 0101 & $x^2 + 1$ & 1101 & $x^3 + x^2 + 1$ \\ 0110 & $x^2 + x$ & 1110 & $x^3 + x^2 + x$ \\ 0111 & $x^2 + x + 1$ & 1111 & $x^3 + x^2 + x+ 1$ \\\hline \end{tabular}\end{split}\]

INPUT:

• G – an element of \(\GF{2^4}\), a list of elements of \(\GF{2^4}\), or a matrix over \(\GF{2^4}\)

OUTPUT:

• A binary string representation of G.

EXAMPLES:

Obtain the binary representation of all elements of \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: S = Set(K); len(S)  # GF(2^4) has this many elements
16
sage: [maes.GF_to_binary(S[i]) for i in range(len(S))]

[0000,
0001,
0010,
0011,
0100,
0101,
0110,
0111,
1000,
1001,
1010,
1011,
1100,
1101,
1110,
1111]

The binary representation of a list of elements belonging to \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: G = [K("x^2 + x + 1"), K("x^3 + x^2"), K("x"), K("x^3 + x + 1"), K("x^3 + x^2 + x + 1"), K("x^2 + x"), K("1"), K("x^2 + x + 1")]
sage: maes.GF_to_binary(G)
01111100001010111111011000010111

The binary representation of a matrix over \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: G = MS([K("x^3 + x^2"), K("x + 1"), K("x^2 + x + 1"), K("x^3 + x^2 + x")]); G

[    x^3 + x^2         x + 1]
[  x^2 + x + 1 x^3 + x^2 + x]
sage: maes.GF_to_binary(G)
1100001101111110
sage: MS = MatrixSpace(K, 2, 4)
sage: G = MS([K("x^2 + x + 1"), K("x^3 + x^2"), K("x"), K("x^3 + x + 1"), K("x^3 + x^2 + x + 1"), K("x^2 + x"), K("1"), K("x^2 + x + 1")]); G

[      x^2 + x + 1         x^3 + x^2                 x       x^3 + x + 1]
[x^3 + x^2 + x + 1           x^2 + x                 1       x^2 + x + 1]
sage: maes.GF_to_binary(G)
01111100001010111111011000010111

GF_to_integer(G)

Return the integer representation of the finite field element G. If G is an element of the finite field \(\GF{2^4}\), then obtain the integer representation of G. If G is a list of elements belonging to \(\GF{2^4}\), obtain the integer representation of each element of the list, and return the result as a list of integers. If G is a matrix with entries over \(\GF{2^4}\), convert each matrix entry to its integer representation, and return the result as a list of integers. The resulting list is obtained by starting from the top-left corner of the matrix, working across left to right, top to bottom. Each element of \(\GF{2^4}\) can be associated with a unique integer according to the following table:

\[\begin{split}\begin{tabular}{ll|ll} \hline integer & $\GF{2^4}$ & integer & $\GF{2^4}$ \\\hline 0 & $0$ & 8 & $x^3$ \\ 1 & $1$ & 9 & $x^3 + 1$ \\ 2 & $x$ & 10 & $x^3 + x$ \\ 3 & $x + 1$ & 11 & $x^3 + x + 1$ \\ 4 & $x^2$ & 12 & $x^3 + x^2$ \\ 5 & $x^2 + 1$ & 13 & $x^3 + x^2 + 1$ \\ 6 & $x^2 + x$ & 14 & $x^3 + x^2 + x$ \\ 7 & $x^2 + x + 1$ & 15 & $x^3 + x^2 + x+ 1$ \\\hline \end{tabular}\end{split}\]

INPUT:

• G – an element of \(\GF{2^4}\), a list of elements belonging to \(\GF{2^4}\), or a matrix over \(\GF{2^4}\)

OUTPUT:

• The integer representation of G.

EXAMPLES:

Obtain the integer representation of all elements of \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: S = Set(K); len(S)  # GF(2^4) has this many elements
16
sage: [maes.GF_to_integer(S[i]) for i in range(len(S))]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

The integer representation of a list of elements belonging to \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: G = [K("x^2 + x + 1"), K("x^3 + x^2"), K("x"), K("x^3 + x + 1"), K("x^3 + x^2 + x + 1"), K("x^2 + x"), K("1"), K("x^2 + x + 1")]
sage: maes.GF_to_integer(G)
[7, 12, 2, 11, 15, 6, 1, 7]

The integer representation of a matrix over \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: G = MS([K("x^3 + x^2"), K("x + 1"), K("x^2 + x + 1"), K("x^3 + x^2 + x")]); G

[    x^3 + x^2         x + 1]
[  x^2 + x + 1 x^3 + x^2 + x]
sage: maes.GF_to_integer(G)
[12, 3, 7, 14]
sage: MS = MatrixSpace(K, 2, 4)
sage: G = MS([K("x^2 + x + 1"), K("x^3 + x^2"), K("x"), K("x^3 + x + 1"), K("x^3 + x^2 + x + 1"), K("x^2 + x"), K("1"), K("x^2 + x + 1")]); G

[      x^2 + x + 1         x^3 + x^2                 x       x^3 + x + 1]
[x^3 + x^2 + x + 1           x^2 + x                 1       x^2 + x + 1]
sage: maes.GF_to_integer(G)
[7, 12, 2, 11, 15, 6, 1, 7]

add_key(block, rkey)

Return the matrix addition of block and rkey. Both block and rkey are \(2 \times 2\) matrices over the finite field \(\GF{2^4}\). This method just return the matrix addition of these two matrices.

INPUT:

• block – a \(2 \times 2\) matrix with entries over \(\GF{2^4}\)

• rkey – a round key; a \(2 \times 2\) matrix with entries over \(\GF{2^4}\)

OUTPUT:

• The matrix addition of block and rkey.

EXAMPLES:

We can work with elements of \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: D = MS([ [K("x^3 + x^2 + x + 1"), K("x^3 + x")], [K("0"), K("x^3 + x^2")] ]); D

[x^3 + x^2 + x + 1           x^3 + x]
[                0         x^3 + x^2]
sage: k = MS([ [K("x^2 + 1"), K("x^3 + x^2 + x + 1")], [K("x + 1"), K("0")] ]); k

[          x^2 + 1 x^3 + x^2 + x + 1]
[            x + 1                 0]
sage: maes.add_key(D, k)

[  x^3 + x   x^2 + 1]
[    x + 1 x^3 + x^2]

Or work with binary strings:

sage: bin = BinaryStrings()
sage: B = bin.encoding("We"); B
0101011101100101
sage: B = MS(maes.binary_to_GF(B)); B

[    x^2 + 1 x^2 + x + 1]
[    x^2 + x     x^2 + 1]
sage: key = bin.encoding("KY"); key
0100101101011001
sage: key = MS(maes.binary_to_GF(key)); key

[        x^2 x^3 + x + 1]
[    x^2 + 1     x^3 + 1]
sage: maes.add_key(B, key)

[        1 x^3 + x^2]
[    x + 1 x^3 + x^2]

We can also work with integers \(n\) such that \(0 \leq n \leq 15\):

sage: N = [2, 3, 5, 7]; N
[2, 3, 5, 7]
sage: key = [9, 11, 13, 15]; key
[9, 11, 13, 15]
sage: N = MS(maes.integer_to_GF(N)); N

[          x       x + 1]
[    x^2 + 1 x^2 + x + 1]
sage: key = MS(maes.integer_to_GF(key)); key

[          x^3 + 1       x^3 + x + 1]
[    x^3 + x^2 + 1 x^3 + x^2 + x + 1]
sage: maes.add_key(N, key)

[x^3 + x + 1         x^3]
[        x^3         x^3]

binary_to_GF(B)

Return a list of elements of \(\GF{2^4}\) that represents the binary string B. The number of bits in B must be greater than zero and a multiple of 4. Each nibble (or 4-bit string) is uniquely associated with an element of \(\GF{2^4}\) as specified by the following table:

\[\begin{split}\begin{tabular}{ll|ll} \hline 4-bit string & $\GF{2^4}$ & 4-bit string & $\GF{2^4}$ \\\hline 0000 & $0$ & 1000 & $x^3$ \\ 0001 & $1$ & 1001 & $x^3 + 1$ \\ 0010 & $x$ & 1010 & $x^3 + x$ \\ 0011 & $x + 1$ & 1011 & $x^3 + x + 1$ \\ 0100 & $x^2$ & 1100 & $x^3 + x^2$ \\ 0101 & $x^2 + 1$ & 1101 & $x^3 + x^2 + 1$ \\ 0110 & $x^2 + x$ & 1110 & $x^3 + x^2 + x$ \\ 0111 & $x^2 + x + 1$ & 1111 & $x^3 + x^2 + x+ 1$ \\\hline \end{tabular}\end{split}\]

INPUT:

• B – a binary string, where the number of bits is positive and a multiple of 4

OUTPUT:

• A list of elements of the finite field \(\GF{2^4}\) that represent the binary string B.

EXAMPLES:

Obtain all the elements of the finite field \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: bin = BinaryStrings()
sage: B = bin("0000000100100011010001010110011110001001101010111100110111101111")
sage: maes.binary_to_GF(B)

[0,
1,
x,
x + 1,
x^2,
x^2 + 1,
x^2 + x,
x^2 + x + 1,
x^3,
x^3 + 1,
x^3 + x,
x^3 + x + 1,
x^3 + x^2,
x^3 + x^2 + 1,
x^3 + x^2 + x,
x^3 + x^2 + x + 1]

binary_to_integer(B)

Return a list of integers representing the binary string B. The number of bits in B must be greater than zero and a multiple of 4. Each nibble (or 4-bit string) is uniquely associated with an integer as specified by the following table:

\[\begin{split}\begin{tabular}{ll|ll} \hline 4-bit string & integer & 4-bit string & integer \\\hline 0000 & 0 & 1000 & 8 \\ 0001 & 1 & 1001 & 9 \\ 0010 & 2 & 1010 & 10 \\ 0011 & 3 & 1011 & 11 \\ 0100 & 4 & 1100 & 12 \\ 0101 & 5 & 1101 & 13 \\ 0110 & 6 & 1110 & 14 \\ 0111 & 7 & 1111 & 15 \\\hline \end{tabular}\end{split}\]

INPUT:

• B – a binary string, where the number of bits is positive and a multiple of 4

OUTPUT:

• A list of integers that represent the binary string B.

EXAMPLES:

Obtain the integer representation of every 4-bit string:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: bin = BinaryStrings()
sage: B = bin("0000000100100011010001010110011110001001101010111100110111101111")
sage: maes.binary_to_integer(B)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

block_length()

Return the block length of Phan’s Mini-AES block cipher. A key in Phan’s Mini-AES is a block of 16 bits. Each nibble of a key can be considered as an element of the finite field \(\GF{2^4}\). Therefore the key consists of four elements from \(\GF{2^4}\).

OUTPUT:

• The block (or key) length in number of bits.

EXAMPLES:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: maes.block_length()
16

decrypt(C, key)

Use Phan’s Mini-AES to decrypt the ciphertext C with the secret key key. Both C and key must be \(2 \times 2\) matrices over the finite field \(\GF{2^4}\). Let \(\gamma\) denote the operation of nibble-sub, \(\pi\) denote shift-row, \(\theta\) denote mix-column, and \(\sigma_{K_i}\) denote add-key with the round key \(K_i\). Then decryption \(D\) using Phan’s Mini-AES is the function composition

\[D = \sigma_{K_0} \circ \gamma^{-1} \circ \pi \circ \theta \circ \sigma_{K_1} \circ \gamma^{-1} \circ \pi \circ \sigma_{K_2}\]

where \(\gamma^{-1}\) is the nibble-sub operation that uses the S-box for decryption, and the order of execution is from right to left.

INPUT:

• C – a ciphertext block; must be a \(2 \times 2\) matrix over the finite field \(\GF{2^4}\)

• key – a secret key for this Mini-AES block cipher; must be a \(2 \times 2\) matrix over the finite field \(\GF{2^4}\)

OUTPUT:

• The plaintext corresponding to C.

EXAMPLES:

We encrypt a plaintext, decrypt the ciphertext, then compare the decrypted plaintext with the original plaintext. Here we work with elements of \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: P = MS([ [K("x^3 + 1"), K("x^2 + x")], [K("x^3 + x^2"), K("x + 1")] ]); P

[  x^3 + 1   x^2 + x]
[x^3 + x^2     x + 1]
sage: key = MS([ [K("x^3 + x^2"), K("x^3 + x^2 + x + 1")], [K("x + 1"), K("0")] ]); key

[        x^3 + x^2 x^3 + x^2 + x + 1]
[            x + 1                 0]
sage: C = maes.encrypt(P, key); C
[            x + 1           x^2 + 1]
[x^3 + x^2 + x + 1               x^2]
sage: plaintxt = maes.decrypt(C, key)
sage: plaintxt; P

[  x^3 + 1   x^2 + x]
[x^3 + x^2     x + 1]

[  x^3 + 1   x^2 + x]
[x^3 + x^2     x + 1]
sage: plaintxt == P
True

But we can also work with binary strings:

sage: bin = BinaryStrings()
sage: P = bin.encoding("de"); P
0110010001100101
sage: P = MS(maes.binary_to_GF(P)); P

[x^2 + x     x^2]
[x^2 + x x^2 + 1]
sage: key = bin.encoding("ke"); key
0110101101100101
sage: key = MS(maes.binary_to_GF(key)); key

[    x^2 + x x^3 + x + 1]
[    x^2 + x     x^2 + 1]
sage: C = maes.encrypt(P, key)
sage: plaintxt = maes.decrypt(C, key)
sage: plaintxt == P
True

Here we work with integers \(n\) such that \(0 \leq n \leq 15\):

sage: P = [3, 5, 7, 14]; P
[3, 5, 7, 14]
sage: key = [2, 6, 7, 8]; key
[2, 6, 7, 8]
sage: P = MS(maes.integer_to_GF(P)); P

[        x + 1       x^2 + 1]
[  x^2 + x + 1 x^3 + x^2 + x]
sage: key = MS(maes.integer_to_GF(key)); key

[          x     x^2 + x]
[x^2 + x + 1         x^3]
sage: C = maes.encrypt(P, key)
sage: plaintxt = maes.decrypt(C, key)
sage: plaintxt == P
True

encrypt(P, key)

Use Phan’s Mini-AES to encrypt the plaintext P with the secret key key. Both P and key must be \(2 \times 2\) matrices over the finite field \(\GF{2^4}\). Let \(\gamma\) denote the operation of nibble-sub, \(\pi\) denote shift-row, \(\theta\) denote mix-column, and \(\sigma_{K_i}\) denote add-key with the round key \(K_i\). Then encryption \(E\) using Phan’s Mini-AES is the function composition

\[E = \sigma_{K_2} \circ \pi \circ \gamma \circ \sigma_{K_1} \circ \theta \circ \pi \circ \gamma \circ \sigma_{K_0}\]

where the order of execution is from right to left. Note that \(\gamma\) is the nibble-sub operation that uses the S-box for encryption.

INPUT:

• P – a plaintext block; must be a \(2 \times 2\) matrix over the finite field \(\GF{2^4}\)

• key – a secret key for this Mini-AES block cipher; must be a \(2 \times 2\) matrix over the finite field \(\GF{2^4}\)

OUTPUT:

• The ciphertext corresponding to P.

EXAMPLES:

Here we work with elements of \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: P = MS([ [K("x^3 + 1"), K("x^2 + x")], [K("x^3 + x^2"), K("x + 1")] ]); P

[  x^3 + 1   x^2 + x]
[x^3 + x^2     x + 1]
sage: key = MS([ [K("x^3 + x^2"), K("x^3 + x^2 + x + 1")], [K("x + 1"), K("0")] ]); key

[        x^3 + x^2 x^3 + x^2 + x + 1]
[            x + 1                 0]
sage: maes.encrypt(P, key)
[            x + 1           x^2 + 1]
[x^3 + x^2 + x + 1               x^2]

But we can also work with binary strings:

sage: bin = BinaryStrings()
sage: P = bin.encoding("de"); P
0110010001100101
sage: P = MS(maes.binary_to_GF(P)); P

[x^2 + x     x^2]
[x^2 + x x^2 + 1]
sage: key = bin.encoding("ke"); key
0110101101100101
sage: key = MS(maes.binary_to_GF(key)); key

[    x^2 + x x^3 + x + 1]
[    x^2 + x     x^2 + 1]
sage: C = maes.encrypt(P, key)
sage: plaintxt = maes.decrypt(C, key)
sage: plaintxt == P
True

Now we work with integers \(n\) such that \(0 \leq n \leq 15\):

sage: P = [1, 5, 8, 12]; P
[1, 5, 8, 12]
sage: key = [5, 9, 15, 0]; key
[5, 9, 15, 0]
sage: P = MS(maes.integer_to_GF(P)); P

[        1   x^2 + 1]
[      x^3 x^3 + x^2]
sage: key = MS(maes.integer_to_GF(key)); key

[          x^2 + 1           x^3 + 1]
[x^3 + x^2 + x + 1                 0]
sage: C = maes.encrypt(P, key)
sage: plaintxt = maes.decrypt(C, key)
sage: plaintxt == P
True

integer_to_GF(N)

Return the finite field representation of N. If \(N\) is an integer such that \(0 \leq N \leq 15\), return the element of \(\GF{2^4}\) that represents N. If N is a list of integers each of which is \(\geq 0\) and \(\leq 15\), then obtain the element of \(\GF{2^4}\) that represents each such integer, and return a list of such finite field representations. Each integer between 0 and 15, inclusive, can be associated with a unique element of \(\GF{2^4}\) according to the following table:

\[\begin{split}\begin{tabular}{ll|ll} \hline integer & $\GF{2^4}$ & integer & $\GF{2^4}$ \\\hline 0 & $0$ & 8 & $x^3$ \\ 1 & $1$ & 9 & $x^3 + 1$ \\ 2 & $x$ & 10 & $x^3 + x$ \\ 3 & $x + 1$ & 11 & $x^3 + x + 1$ \\ 4 & $x^2$ & 12 & $x^3 + x^2$ \\ 5 & $x^2 + 1$ & 13 & $x^3 + x^2 + 1$ \\ 6 & $x^2 + x$ & 14 & $x^3 + x^2 + x$ \\ 7 & $x^2 + x + 1$ & 15 & $x^3 + x^2 + x+ 1$ \\\hline \end{tabular}\end{split}\]

INPUT:

• N – a non-negative integer less than or equal to 15, or a list of such integers

OUTPUT:

• Elements of the finite field \(\GF{2^4}\).

EXAMPLES:

Obtain the element of \(\GF{2^4}\) representing an integer \(n\), where \(0 \leq n \leq 15\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: maes.integer_to_GF(0)
0
sage: maes.integer_to_GF(2)
x
sage: maes.integer_to_GF(7)
x^2 + x + 1

Obtain the finite field elements corresponding to all non-negative integers less than or equal to 15:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: lst = [n for n in range(16)]; lst
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
sage: maes.integer_to_GF(lst)

[0,
1,
x,
x + 1,
x^2,
x^2 + 1,
x^2 + x,
x^2 + x + 1,
x^3,
x^3 + 1,
x^3 + x,
x^3 + x + 1,
x^3 + x^2,
x^3 + x^2 + 1,
x^3 + x^2 + x,
x^3 + x^2 + x + 1]

integer_to_binary(N)

Return the binary representation of N. If \(N\) is an integer such that \(0 \leq N \leq 15\), return the binary representation of N. If N is a list of integers each of which is \(\geq 0\) and \(\leq 15\), then obtain the binary representation of each integer, and concatenate the individual binary representations into a single binary string. Each integer between 0 and 15, inclusive, can be associated with a unique 4-bit string according to the following table:

\[\begin{split}\begin{tabular}{ll|ll} \hline 4-bit string & integer & 4-bit string & integer \\\hline 0000 & 0 & 1000 & 8 \\ 0001 & 1 & 1001 & 9 \\ 0010 & 2 & 1010 & 10 \\ 0011 & 3 & 1011 & 11 \\ 0100 & 4 & 1100 & 12 \\ 0101 & 5 & 1101 & 13 \\ 0110 & 6 & 1110 & 14 \\ 0111 & 7 & 1111 & 15 \\\hline \end{tabular}\end{split}\]

INPUT:

• N – a non-negative integer less than or equal to 15, or a list of such integers

OUTPUT:

• A binary string representing N.

EXAMPLES:

The binary representations of all integers between 0 and 15, inclusive:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: lst = [n for n in range(16)]; lst
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
sage: maes.integer_to_binary(lst)
0000000100100011010001010110011110001001101010111100110111101111

The binary representation of an integer between 0 and 15, inclusive:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: maes.integer_to_binary(3)
0011
sage: maes.integer_to_binary(5)
0101
sage: maes.integer_to_binary(7)
0111

mix_column(block)

Return the matrix multiplication of block with a constant matrix. The constant matrix is

\[\begin{split}\begin{bmatrix} x + 1 & x \\ x & x + 1 \end{bmatrix}\end{split}\]

If the input block is

\[\begin{split}\begin{bmatrix} c_0 & c_2 \\ c_1 & c_3 \end{bmatrix}\end{split}\]

then the output block is

\[\begin{split}\begin{bmatrix} d_0 & d_2 \\ d_1 & d_3 \end{bmatrix} = \begin{bmatrix} x + 1 & x \\ x & x + 1 \end{bmatrix} \begin{bmatrix} c_0 & c_2 \\ c_1 & c_3 \end{bmatrix}\end{split}\]

INPUT:

• block – a \(2 \times 2\) matrix with entries over \(\GF{2^4}\)

OUTPUT:

• A \(2 \times 2\) matrix resulting from multiplying the above constant matrix with the input matrix block.

EXAMPLES:

Here we work with elements of \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: mat = MS([ [K("x^2 + x + 1"), K("x^3 + x^2 + 1")], [K("x^3"), K("x")] ])
sage: maes.mix_column(mat)

[          x^3 + x                 0]
[          x^2 + 1 x^3 + x^2 + x + 1]

Multiplying by the identity matrix should leave the constant matrix unchanged:

sage: eye = MS([ [K("1"), K("0")], [K("0"), K("1")] ])
sage: maes.mix_column(eye)

[x + 1     x]
[    x x + 1]

We can also work with binary strings:

sage: bin = BinaryStrings()
sage: B = bin.encoding("rT"); B
0111001001010100
sage: B = MS(maes.binary_to_GF(B)); B

[x^2 + x + 1           x]
[    x^2 + 1         x^2]
sage: maes.mix_column(B)

[        x + 1 x^3 + x^2 + x]
[            1           x^3]

We can also work with integers \(n\) such that \(0 \leq n \leq 15\):

sage: P = [10, 5, 2, 7]; P
[10, 5, 2, 7]
sage: P = MS(maes.integer_to_GF(P)); P

[    x^3 + x     x^2 + 1]
[          x x^2 + x + 1]
sage: maes.mix_column(P)

[x^3 + 1       1]
[      1   x + 1]

nibble_sub(block, algorithm='encrypt')

Substitute a nibble (or a block of 4 bits) using the following S-box:

\[\begin{split}\begin{tabular}{ll|ll} \hline Input & Output & Input & Output \\\hline 0000 & 1110 & 1000 & 0011 \\ 0001 & 0100 & 1001 & 1010 \\ 0010 & 1101 & 1010 & 0110 \\ 0011 & 0001 & 1011 & 1100 \\ 0100 & 0010 & 1100 & 0101 \\ 0101 & 1111 & 1101 & 1001 \\ 0110 & 1011 & 1110 & 0000 \\ 0111 & 1000 & 1111 & 0111 \\\hline \end{tabular}\end{split}\]

The values in the above S-box are taken from the first row of the first S-box of the Data Encryption Standard (DES). Each nibble can be thought of as an element of the finite field \(\GF{2^4}\) of 16 elements. Thus in terms of \(\GF{2^4}\), the S-box can also be specified as:

\[\begin{split}\begin{tabular}{ll} \hline Input & Output \\\hline $0$ & $x^3 + x^2 + x$ \\ $1$ & $x^2$ \\ $x$ & $x^3 + x^2 + 1$ \\ $x + 1$ & $1$ \\ $x^2$ & $x$ \\ $x^2 + 1$ & $x^3 + x^2 + x + 1$ \\ $x^2 + x$ & $x^3 + x + 1$ \\ $x^2 + x + 1$ & $x^3$ \\ $x^3$ & $x + 1$ \\ $x^3 + 1$ & $x^3 + x$ \\ $x^3 + x$ & $x^2 + x$ \\ $x^3 + x + 1$ & $x^3 + x^2$ \\ $x^3 + x^2$ & $x^2 + 1$ \\ $x^3 + x^2 + 1$ & $x^3 + 1$ \\ $x^3 + x^2 + x$ & $0$ \\ $x^3 + x^2 + x + 1$ & $x^2 + x + 1$ \\\hline \end{tabular}\end{split}\]

Note that the above S-box is used for encryption. The S-box for decryption is obtained from the above S-box by reversing the role of the Input and Output columns. Thus the previous Input column for encryption now becomes the Output column for decryption, and the previous Output column for encryption is now the Input column for decryption. The S-box used for decryption can be specified as:

\[\begin{split}\begin{tabular}{ll} \hline Input & Output \\\hline $0$ & $x^3 + x^2 + x$ \\ $1$ & $x + 1$ \\ $x$ & $x^2$ \\ $x + 1$ & $x^3$ \\ $x^2$ & $1$ \\ $x^2 + 1$ & $x^3 + x^2$ \\ $x^2 + x$ & $x^3 + x$ \\ $x^2 + x + 1$ & $x^3 + x^2 + x + 1$ \\ $x^3$ & $x^2 + x + 1$ \\ $x^3 + 1$ & $x^3 + x^2 + 1$ \\ $x^3 + x$ & $x^3 + 1$ \\ $x^3 + x + 1$ & $x^2 + x$ \\ $x^3 + x^2$ & $x^3 + x + 1$ \\ $x^3 + x^2 + 1$ & $x$ \\ $x^3 + x^2 + x$ & $0$ \\ $x^3 + x^2 + x + 1$ & $x^2 + 1$ \\\hline \end{tabular}\end{split}\]

INPUT:

• block – a \(2 \times 2\) matrix with entries over \(\GF{2^4}\)

• algorithm – (default: "encrypt") a string; a flag to signify whether this nibble-sub operation is used for encryption or decryption. The encryption flag is "encrypt" and the decryption flag is "decrypt".

OUTPUT:

• A \(2 \times 2\) matrix resulting from applying an S-box on entries of the \(2 \times 2\) matrix block.

EXAMPLES:

Here we work with elements of the finite field \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: mat = MS([[K("x^3 + x^2 + x + 1"), K("0")], [K("x^2 + x + 1"), K("x^3 + x")]])
sage: maes.nibble_sub(mat, algorithm="encrypt")

[  x^2 + x + 1 x^3 + x^2 + x]
[          x^3       x^2 + x]

But we can also work with binary strings:

sage: bin = BinaryStrings()
sage: B = bin.encoding("bi"); B
0110001001101001
sage: B = MS(maes.binary_to_GF(B)); B

[x^2 + x       x]
[x^2 + x x^3 + 1]
sage: maes.nibble_sub(B, algorithm="encrypt")

[  x^3 + x + 1 x^3 + x^2 + 1]
[  x^3 + x + 1       x^3 + x]
sage: maes.nibble_sub(B, algorithm="decrypt")

[      x^3 + x           x^2]
[      x^3 + x x^3 + x^2 + 1]

Here we work with integers \(n\) such that \(0 \leq n \leq 15\):

sage: P = [2, 6, 8, 14]; P
[2, 6, 8, 14]
sage: P = MS(maes.integer_to_GF(P)); P

[            x       x^2 + x]
[          x^3 x^3 + x^2 + x]
sage: maes.nibble_sub(P, algorithm="encrypt")

[x^3 + x^2 + 1   x^3 + x + 1]
[        x + 1             0]
sage: maes.nibble_sub(P, algorithm="decrypt")

[        x^2     x^3 + x]
[x^2 + x + 1           0]

random_key()

A random key within the key space of this Mini-AES block cipher. Like the AES, Phan’s Mini-AES is a symmetric-key block cipher. A Mini-AES key is a block of 16 bits, or a \(2 \times 2\) matrix with entries over the finite field \(\GF{2^4}\). Thus the number of possible keys is \(2^{16} = 16^4\).

OUTPUT:

• A \(2 \times 2\) matrix over the finite field \(\GF{2^4}\), used as a secret key for this Mini-AES block cipher.

EXAMPLES:

Each nibble of a key is an element of the finite field \(\GF{2^4}\):

sage: K = FiniteField(16, "x")
sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: key = maes.random_key()
sage: [key[i][j] in K for i in range(key.nrows()) for j in range(key.ncols())]
[True, True, True, True]

Generate a random key, then perform encryption and decryption using that key:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: key = maes.random_key()
sage: P = MS.random_element()
sage: C = maes.encrypt(P, key)
sage: plaintxt = maes.decrypt(C, key)
sage: plaintxt == P
True

round_key(key, n)

Return the round key for round n. Phan’s Mini-AES is defined to have two rounds. The round key \(K_0\) is generated and used prior to the first round, with round keys \(K_1\) and \(K_2\) being used in rounds 1 and 2 respectively. In total, there are three round keys, each generated from the secret key key.

INPUT:

• key – the secret key

• n – non-negative integer; the round number

OUTPUT:

• The \(n\)-th round key.

EXAMPLES:

Obtaining the round keys from the secret key:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: key = MS([ [K("x^3 + x^2"), K("x^3 + x^2 + x + 1")], [K("x + 1"), K("0")] ])
sage: maes.round_key(key, 0)

[        x^3 + x^2 x^3 + x^2 + x + 1]
[            x + 1                 0]
sage: key

[        x^3 + x^2 x^3 + x^2 + x + 1]
[            x + 1                 0]
sage: maes.round_key(key, 1)
[x^3 + x x^2 + x]
[x^3 + 1 x^2 + x]
sage: maes.round_key(key, 2)

[  x^2 + 1   x^3 + x]
[x^3 + x^2 x^3 + x^2]

sbox()

Return the S-box of Mini-AES.

EXAMPLES:

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: maes.sbox()
(14, 4, 13, 1, 2, 15, 11, 8, 3, 10, 6, 12, 5, 9, 0, 7)

shift_row(block)

Rotate each row of block to the left by different nibble amounts. The first or zero-th row is left unchanged, while the second or row one is rotated left by one nibble. This has the effect of only interchanging the nibbles in the second row. Let \(b_0, b_1, b_2, b_3\) be four nibbles arranged as the following \(2 \times 2\) matrix

\[\begin{split}\begin{bmatrix} b_0 & b_2 \\ b_1 & b_3 \end{bmatrix}\end{split}\]

Then the operation of shift-row is the mapping

\[\begin{split}\begin{bmatrix} b_0 & b_2 \\ b_1 & b_3 \end{bmatrix} \longmapsto \begin{bmatrix} b_0 & b_2 \\ b_3 & b_1 \end{bmatrix}\end{split}\]

INPUT:

• block – a \(2 \times 2\) matrix with entries over \(\GF{2^4}\)

OUTPUT:

• A \(2 \times 2\) matrix resulting from applying shift-row on block.

EXAMPLES:

Here we work with elements of the finite field \(\GF{2^4}\):

sage: from sage.crypto.block_cipher.miniaes import MiniAES
sage: maes = MiniAES()
sage: K = FiniteField(16, "x")
sage: MS = MatrixSpace(K, 2, 2)
sage: mat = MS([[K("x^3 + x^2 + x + 1"), K("0")], [K("x^2 + x + 1"), K("x^3 + x")]])
sage: maes.shift_row(mat)

[x^3 + x^2 + x + 1                 0]
[          x^3 + x       x^2 + x + 1]
sage: mat

[x^3 + x^2 + x + 1                 0]
[      x^2 + x + 1           x^3 + x]

But we can also work with binary strings:

sage: bin = BinaryStrings()
sage: B = bin.encoding("Qt"); B
0101000101110100
sage: B = MS(maes.binary_to_GF(B)); B

[    x^2 + 1           1]
[x^2 + x + 1         x^2]
sage: maes.shift_row(B)

[    x^2 + 1           1]
[        x^2 x^2 + x + 1]

Here we work with integers \(n\) such that \(0 \leq n \leq 15\):

sage: P = [3, 6, 9, 12]; P
[3, 6, 9, 12]
sage: P = MS(maes.integer_to_GF(P)); P

[    x + 1   x^2 + x]
[  x^3 + 1 x^3 + x^2]
sage: maes.shift_row(P)

[    x + 1   x^2 + x]
[x^3 + x^2   x^3 + 1]