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: sage.structure.sage_object.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       x^2 + x]
[x^3 + x^2 + x       x^3 + 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
10001000101001101111000001111000010011001110110101000111011011010101001011101111101011001110011100100011101100101010100010100111110110011001010001000111011011010010000011000110001100000111000011100110101111000000001110001001
sage: plaintxt = maes(C, key, algorithm="decrypt")
sage: plaintxt == P
True

Now we work with integers n such that 0n15:

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
1100100000100011111001010101010101011011100111110001000011100001
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 F24, then obtain the binary representation of G. If G is a list of elements belonging to F24, 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 F24, 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 F24 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 F24, a list of elements of F24, or a matrix over F24

OUTPUT:

  • A binary string representation of G.

EXAMPLES:

Obtain the binary representation of all elements of F24:

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 F24:

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 F24:

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 F24, then obtain the integer representation of G. If G is a list of elements belonging to F24, 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 F24, 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 F24 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 F24, a list of elements belonging to F24, or a matrix over F24

OUTPUT:

  • The integer representation of G.

EXAMPLES:

Obtain the integer representation of all elements of F24:

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 F24:

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 F24:

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 2×2 matrices over the finite field F24. This method just return the matrix addition of these two matrices.

INPUT:

  • block – a 2×2 matrix with entries over F24

  • rkey – a round key; a 2×2 matrix with entries over F24

OUTPUT:

  • The matrix addition of block and rkey.

EXAMPLES:

We can work with elements of F24:

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 0n15:

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 F24 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 F24 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 F24 that represent the binary string B.

EXAMPLES:

Obtain all the elements of the finite field F24:

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 F24. Therefore the key consists of four elements from F24.

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 2×2 matrices over the finite field F24. Let γ denote the operation of nibble-sub, π denote shift-row, θ denote mix-column, and σKi denote add-key with the round key Ki. Then decryption D using Phan’s Mini-AES is the function composition

D=σK0γ1πθσK1γ1πσK2

where γ1 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 2×2 matrix over the finite field F24

  • key – a secret key for this Mini-AES block cipher; must be a 2×2 matrix over the finite field F24

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 F24:

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^2 + x + 1   x^3 + x^2]
[          x     x^2 + x]
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 0n15:

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 2×2 matrices over the finite field F24. Let γ denote the operation of nibble-sub, π denote shift-row, θ denote mix-column, and σKi denote add-key with the round key Ki. Then encryption E using Phan’s Mini-AES is the function composition

E=σK2πγσK1θπγσK0

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

INPUT:

  • P – a plaintext block; must be a 2×2 matrix over the finite field F24

  • key – a secret key for this Mini-AES block cipher; must be a 2×2 matrix over the finite field F24

OUTPUT:

  • The ciphertext corresponding to P.

EXAMPLES:

Here we work with elements of F24:

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

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 0n15:

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 0N15, return the element of F24 that represents N. If N is a list of integers each of which is 0 and 15, then obtain the element of F24 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 F24 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 F24.

EXAMPLES:

Obtain the element of F24 representing an integer n, where 0n15:

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 0N15, return the binary representation of N. If N is a list of integers each of which is 0 and 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

[x+1xxx+1]

If the input block is

[c0c2c1c3]

then the output block is

[d0d2d1d3]=[x+1xxx+1][c0c2c1c3]

INPUT:

  • block – a 2×2 matrix with entries over F24

OUTPUT:

  • A 2×2 matrix resulting from multiplying the above constant matrix with the input matrix block.

EXAMPLES:

Here we work with elements of F24:

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 0n15:

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 F24 of 16 elements. Thus in terms of F24, 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 2×2 matrix with entries over F24

  • 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 2×2 matrix resulting from applying an S-box on entries of the 2×2 matrix block.

EXAMPLES:

Here we work with elements of the finite field F24:

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 0n15:

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 2×2 matrix with entries over the finite field F24. Thus the number of possible keys is 216=164.

OUTPUT:

  • A 2×2 matrix over the finite field F24, used as a secret key for this Mini-AES block cipher.

EXAMPLES:

Each nibble of a key is an element of the finite field F24:

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 K0 is generated and used prior to the first round, with round keys K1 and K2 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 + 1 x^3 + x^2 + x + 1]
[                0 x^3 + x^2 + x + 1]
sage: maes.round_key(key, 2)

[x^2 + x x^3 + 1]
[x^2 + x x^2 + x]
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 b0,b1,b2,b3 be four nibbles arranged as the following 2×2 matrix

[b0b2b1b3]

Then the operation of shift-row is the mapping

[b0b2b1b3][b0b2b3b1]

INPUT:

  • block – a 2×2 matrix with entries over F24

OUTPUT:

  • A 2×2 matrix resulting from applying shift-row on block.

EXAMPLES:

Here we work with elements of the finite field F24:

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 0n15:

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]