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r""" 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 """
########################################################################### # Copyright (c) 2009 Minh Van Nguyen <nguyenminh2@gmail.com> # # This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # http://www.gnu.org/licenses/ ###########################################################################
r""" 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 :mod:`SR <sage.crypto.mq.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 <BLANKLINE> [ 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 <BLANKLINE> [ x^3 + x^2 x^3 + x] [x^3 + x^2 + x x^2 + x + 1] sage: C = maes.encrypt(P, key); C <BLANKLINE> [ x x^2 + x] [x^3 + x^2 + x x^3 + x]
Decrypt the result::
sage: plaintxt = maes.decrypt(C, key) sage: plaintxt; P <BLANKLINE> [ x^3 + x x^2 + 1] [ x^2 + x x^3 + x^2] <BLANKLINE> [ 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 `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 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 """
r""" A simplified variant of the Advanced Encryption Standard (AES).
EXAMPLES::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES(); maes Mini-AES block cipher with 16-bit keys 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 """ # the S-box for encryption # the S-box for decryption # nibble to finite field element B("0001"): K("1"), B("0010"): K("x"), B("0011"): K("x + 1"), B("0100"): K("x^2"), B("0101"): K("x^2 + 1"), B("0110"): K("x^2 + x"), B("0111"): K("x^2 + x + 1"), B("1000"): K("x^3"), B("1001"): K("x^3 + 1"), B("1010"): K("x^3 + x"), B("1011"): K("x^3 + x + 1"), B("1100"): K("x^3 + x^2"), B("1101"): K("x^3 + x^2 + 1"), B("1110"): K("x^3 + x^2 + x"), B("1111"): K("x^3 + x^2 + x+ 1") } # nibble to integer B("0001"): Integer(1), B("0010"): Integer(2), B("0011"): Integer(3), B("0100"): Integer(4), B("0101"): Integer(5), B("0110"): Integer(6), B("0111"): Integer(7), B("1000"): Integer(8), B("1001"): Integer(9), B("1010"): Integer(10), B("1011"): Integer(11), B("1100"): Integer(12), B("1101"): Integer(13), B("1110"): Integer(14), B("1111"): Integer(15) } # finite field element to nibble K("1"): B("0001"), K("x"): B("0010"), K("x + 1"): B("0011"), K("x^2"): B("0100"), K("x^2 + 1"): B("0101"), K("x^2 + x"): B("0110"), K("x^2 + x + 1"): B("0111"), K("x^3"): B("1000"), K("x^3 + 1"): B("1001"), K("x^3 + x"): B("1010"), K("x^3 + x + 1"): B("1011"), K("x^3 + x^2"): B("1100"), K("x^3 + x^2 + 1"): B("1101"), K("x^3 + x^2 + x"): B("1110"), K("x^3 + x^2 + x+ 1"): B("1111") } # finite field element to integer K("1"): Integer(1), K("x"): Integer(2), K("x + 1"): Integer(3), K("x^2"): Integer(4), K("x^2 + 1"): Integer(5), K("x^2 + x"): Integer(6), K("x^2 + x + 1"): Integer(7), K("x^3"): Integer(8), K("x^3 + 1"): Integer(9), K("x^3 + x"): Integer(10), K("x^3 + x + 1"): Integer(11), K("x^3 + x^2"): Integer(12), K("x^3 + x^2 + 1"): Integer(13), K("x^3 + x^2 + x"): Integer(14), K("x^3 + x^2 + x+ 1"): Integer(15) } # integer to nibble Integer(1): B("0001"), Integer(2): B("0010"), Integer(3): B("0011"), Integer(4): B("0100"), Integer(5): B("0101"), Integer(6): B("0110"), Integer(7): B("0111"), Integer(8): B("1000"), Integer(9): B("1001"), Integer(10): B("1010"), Integer(11): B("1011"), Integer(12): B("1100"), Integer(13): B("1101"), Integer(14): B("1110"), Integer(15): B("1111") } # integer to finite field element Integer(1): K("1"), Integer(2): K("x"), Integer(3): K("x + 1"), Integer(4): K("x^2"), Integer(5): K("x^2 + 1"), Integer(6): K("x^2 + x"), Integer(7): K("x^2 + x + 1"), Integer(8): K("x^3"), Integer(9): K("x^3 + 1"), Integer(10): K("x^3 + x"), Integer(11): K("x^3 + x + 1"), Integer(12): K("x^3 + x^2"), Integer(13): K("x^3 + x^2 + 1"), Integer(14): K("x^3 + x^2 + x"), Integer(15): K("x^3 + x^2 + x+ 1") }
r""" Apply Mini-AES encryption or decryption on the binary string ``B`` using the key ``key``. 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 16. The number of 16 bits is evenly divided into four nibbles. Hence 16 bits can be conveniently represented as a `2 \times 2` matrix block for encryption or decryption.
- ``key`` -- a secret key; this must be a 16-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::
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
TESTS:
The binary string ``B`` must be non-empty and the number of bits must be a multiple of 16::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes("B", "key") Traceback (most recent call last): ... TypeError: input B must be a non-empty binary string with number of bits a multiple of 16 sage: bin = BinaryStrings() sage: B = bin.encoding("A") sage: maes(B, "key") Traceback (most recent call last): ... ValueError: the number of bits in the binary string B must be positive and a multiple of 16
The secret key ``key`` must be a 16-bit binary string::
sage: B = bin.encoding("ABCD") sage: maes(B, "key") Traceback (most recent call last): ... TypeError: secret key must be a 16-bit binary string sage: key = bin.encoding("K") sage: maes(B, key) Traceback (most recent call last): ... ValueError: secret key must be a 16-bit binary string
The value for ``algorithm`` must be either ``"encrypt"`` or ``"decrypt"``::
sage: B = bin.encoding("ABCD") sage: key = bin.encoding("KE") sage: maes(B, key, algorithm="ABC") Traceback (most recent call last): ... ValueError: algorithm must be either 'encrypt' or 'decrypt' sage: maes(B, key, algorithm="e") Traceback (most recent call last): ... ValueError: algorithm must be either 'encrypt' or 'decrypt' sage: maes(B, key, algorithm="d") Traceback (most recent call last): ... ValueError: algorithm must be either 'encrypt' or 'decrypt' """
# encrypt each 16-bit block in succession # here 16 is the number of bits per encryption block # decrypt each 16-bit block in succession # here 16 is the number of bits per encryption block else:
r""" Compare ``self`` with ``other``.
Mini-AES objects are the same if they have the same key size and the same S-boxes.
EXAMPLES::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: m = MiniAES() sage: m == loads(dumps(m)) True """ (self._sboxE == other._sboxE) and (self._sboxD == other._sboxD) )
r""" Return the string representation of self.
EXAMPLES::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: m = MiniAES(); m Mini-AES block cipher with 16-bit keys """
r""" 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 <BLANKLINE> [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 <BLANKLINE> [ x^2 + 1 x^3 + x^2 + x + 1] [ x + 1 0] sage: maes.add_key(D, k) <BLANKLINE> [ 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 <BLANKLINE> [ 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 <BLANKLINE> [ x^2 x^3 + x + 1] [ x^2 + 1 x^3 + 1] sage: maes.add_key(B, key) <BLANKLINE> [ 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 <BLANKLINE> [ x x + 1] [ x^2 + 1 x^2 + x + 1] sage: key = MS(maes.integer_to_GF(key)); key <BLANKLINE> [ x^3 + 1 x^3 + x + 1] [ x^3 + x^2 + 1 x^3 + x^2 + x + 1] sage: maes.add_key(N, key) <BLANKLINE> [x^3 + x + 1 x^3] [ x^3 x^3]
TESTS:
The input block and key must each be a matrix::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: K = FiniteField(16, "x") sage: MSB = MatrixSpace(K, 2, 2) sage: B = MSB([ [K("x^3 + 1"), K("x^2 + x")], [K("x^3 + x^2"), K("x + 1")] ]) sage: maes.add_key(B, "key") Traceback (most recent call last): ... TypeError: round key must be a 2 x 2 matrix over GF(16) sage: maes.add_key("block", "key") Traceback (most recent call last): ... TypeError: input block must be a 2 x 2 matrix over GF(16)
In addition, the dimensions of the input matrices must each be `2 \times 2`::
sage: MSB = MatrixSpace(K, 1, 2) sage: B = MSB([ [K("x^3 + 1"), K("x^2 + x")] ]) sage: maes.add_key(B, "key") Traceback (most recent call last): ... TypeError: input block must be a 2 x 2 matrix over GF(16) sage: MSB = MatrixSpace(K, 2, 2) sage: B = MSB([ [K("x^3 + 1"), K("x^2 + x")], [K("x^3 + x^2"), K("x + 1")] ]) sage: MSK = MatrixSpace(K, 1, 2) sage: key = MSK([ [K("x^3 + x^2"), K("x^3 + x^2 + x + 1")]]) sage: maes.add_key(B, key) Traceback (most recent call last): ... TypeError: round key must be a 2 x 2 matrix over GF(16) """ not (block.base_ring().order() == 16 and block.base_ring().is_field()):
not (rkey.base_ring().order() == 16 and rkey.base_ring().is_field()):
r""" 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 """
r""" 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
.. MATH::
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 <BLANKLINE> [ 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 <BLANKLINE> [ x^3 + x^2 x^3 + x^2 + x + 1] [ x + 1 0] sage: C = maes.encrypt(P, key); C <BLANKLINE> [x^2 + x + 1 x^3 + x^2] [ x x^2 + x] sage: plaintxt = maes.decrypt(C, key) sage: plaintxt; P <BLANKLINE> [ x^3 + 1 x^2 + x] [x^3 + x^2 x + 1] <BLANKLINE> [ 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 <BLANKLINE> [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 <BLANKLINE> [ 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 <BLANKLINE> [ x + 1 x^2 + 1] [ x^2 + x + 1 x^3 + x^2 + x] sage: key = MS(maes.integer_to_GF(key)); key <BLANKLINE> [ 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
TESTS:
The input block must be a matrix::
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.decrypt("C", key) Traceback (most recent call last): ... TypeError: ciphertext block must be a 2 x 2 matrix over GF(16) sage: C = MS([ [K("x^3 + 1"), K("x^2 + x")], [K("x^3 + x^2"), K("x + 1")] ]) sage: maes.decrypt(C, "key") Traceback (most recent call last): ... TypeError: secret key must be a 2 x 2 matrix over GF(16)
In addition, the dimensions of the input matrices must be `2 \times 2`::
sage: MS = MatrixSpace(K, 1, 2) sage: C = MS([ [K("x^3 + 1"), K("x^2 + x")]]) sage: maes.decrypt(C, "key") Traceback (most recent call last): ... TypeError: ciphertext block must be a 2 x 2 matrix over GF(16) sage: MSC = MatrixSpace(K, 2, 2) sage: C = MSC([ [K("x^3 + 1"), K("x^2 + x")], [K("x^3 + x^2"), K("x + 1")] ]) sage: MSK = MatrixSpace(K, 1, 2) sage: key = MSK([ [K("x^3 + x^2"), K("x^3 + x^2 + x + 1")]]) sage: maes.decrypt(C, key) Traceback (most recent call last): ... TypeError: secret key must be a 2 x 2 matrix over GF(16) """ not (C.base_ring().order() == 16 and C.base_ring().is_field()): not (key.base_ring().order() == 16 and key.base_ring().is_field()):
# pre-compute the round keys
# now proceed with decrypting the ciphertext
# undo the result of round 2
# undo the result of round 1
# undo the result of round 0
r""" 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
.. MATH::
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 <BLANKLINE> [ 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 <BLANKLINE> [ x^3 + x^2 x^3 + x^2 + x + 1] [ x + 1 0] sage: maes.encrypt(P, key) <BLANKLINE> [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 <BLANKLINE> [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 <BLANKLINE> [ 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 <BLANKLINE> [ 1 x^2 + 1] [ x^3 x^3 + x^2] sage: key = MS(maes.integer_to_GF(key)); key <BLANKLINE> [ 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
TESTS:
The input block must be a matrix::
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.encrypt("P", key) Traceback (most recent call last): ... TypeError: plaintext block must be a 2 x 2 matrix over GF(16) sage: P = MS([ [K("x^3 + 1"), K("x^2 + x")], [K("x^3 + x^2"), K("x + 1")] ]) sage: maes.encrypt(P, "key") Traceback (most recent call last): ... TypeError: secret key must be a 2 x 2 matrix over GF(16)
In addition, the dimensions of the input matrices must be `2 \times 2`::
sage: MS = MatrixSpace(K, 1, 2) sage: P = MS([ [K("x^3 + 1"), K("x^2 + x")]]) sage: maes.encrypt(P, "key") Traceback (most recent call last): ... TypeError: plaintext block must be a 2 x 2 matrix over GF(16) sage: MSP = MatrixSpace(K, 2, 2) sage: P = MSP([ [K("x^3 + 1"), K("x^2 + x")], [K("x^3 + x^2"), K("x + 1")] ]) sage: MSK = MatrixSpace(K, 1, 2) sage: key = MSK([ [K("x^3 + x^2"), K("x^3 + x^2 + x + 1")]]) sage: maes.encrypt(P, key) Traceback (most recent call last): ... TypeError: secret key must be a 2 x 2 matrix over GF(16) """ not (P.base_ring().order() == 16 and P.base_ring().is_field()): not (key.base_ring().order() == 16 and key.base_ring().is_field()):
# pre-compute the round keys
# now proceed with encrypting the plaintext
# round 0
# round 1
# round 2
r""" Return the matrix multiplication of ``block`` with a constant matrix. The constant matrix is
.. MATH::
\begin{bmatrix} x + 1 & x \\ x & x + 1 \end{bmatrix}
If the input block is
.. MATH::
\begin{bmatrix} c_0 & c_2 \\ c_1 & c_3 \end{bmatrix}
then the output block is
.. MATH::
\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}
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) <BLANKLINE> [ 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) <BLANKLINE> [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 <BLANKLINE> [x^2 + x + 1 x] [ x^2 + 1 x^2] sage: maes.mix_column(B) <BLANKLINE> [ 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 <BLANKLINE> [ x^3 + x x^2 + 1] [ x x^2 + x + 1] sage: maes.mix_column(P) <BLANKLINE> [x^3 + 1 1] [ 1 x + 1]
TESTS:
The input block must be a matrix::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes.mix_column("mat") Traceback (most recent call last): ... TypeError: input block must be a 2 x 2 matrix over GF(16)
In addition, the dimensions of the input matrix must be `2 \times 2`::
sage: K = FiniteField(16, "x") sage: MS = MatrixSpace(K, 1, 2) sage: mat = MS([[K("x^3 + x^2 + x + 1"), K("0")]]) sage: maes.mix_column(mat) Traceback (most recent call last): ... TypeError: input block must be a 2 x 2 matrix over GF(16) """ not (block.base_ring().order() == 16 and block.base_ring().is_field()):
[K("x"), K("x + 1")] ] )
r""" Substitute a nibble (or a block of 4 bits) using the following S-box:
.. MATH::
\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}
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:
.. MATH::
\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}
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:
.. MATH::
\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}
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") <BLANKLINE> [ 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 <BLANKLINE> [x^2 + x x] [x^2 + x x^3 + 1] sage: maes.nibble_sub(B, algorithm="encrypt") <BLANKLINE> [ x^3 + x + 1 x^3 + x^2 + 1] [ x^3 + x + 1 x^3 + x] sage: maes.nibble_sub(B, algorithm="decrypt") <BLANKLINE> [ 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 <BLANKLINE> [ x x^2 + x] [ x^3 x^3 + x^2 + x] sage: maes.nibble_sub(P, algorithm="encrypt") <BLANKLINE> [x^3 + x^2 + 1 x^3 + x + 1] [ x + 1 0] sage: maes.nibble_sub(P, algorithm="decrypt") <BLANKLINE> [ x^2 x^3 + x] [x^2 + x + 1 0]
TESTS:
The input block must be a matrix::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes.nibble_sub("mat") Traceback (most recent call last): ... TypeError: input block must be a 2 x 2 matrix over GF(16)
In addition, the dimensions of the input matrix must be `2 \times 2`::
sage: K = FiniteField(16, "x") sage: MS = MatrixSpace(K, 1, 2) sage: mat = MS([[K("x^3 + x^2 + x + 1"), K("0")]]) sage: maes.nibble_sub(mat) Traceback (most recent call last): ... TypeError: input block must be a 2 x 2 matrix over GF(16)
The value for the option ``algorithm`` must be either the string ``"encrypt"`` or ``"decrypt"``::
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="abc") Traceback (most recent call last): ... ValueError: the algorithm for nibble-sub must be either 'encrypt' or 'decrypt' sage: maes.nibble_sub(mat, algorithm="e") Traceback (most recent call last): ... ValueError: the algorithm for nibble-sub must be either 'encrypt' or 'decrypt' sage: maes.nibble_sub(mat, algorithm="d") Traceback (most recent call last): ... ValueError: the algorithm for nibble-sub must be either 'encrypt' or 'decrypt' """ not (block.base_ring().order() == 16 and block.base_ring().is_field()):
# get the integer representation of each GF(2^4) element # in the input matrix block # Now run each resulting integer through the S-box for # encryption. Then convert the result output by the S-box # to an element of GF(2^4). # Now run each resulting integer through the S-box for # decryption. Then convert the result output by the S-box # to an element of GF(2^4). else:
r""" 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 """
r""" 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) <BLANKLINE> [ x^3 + x^2 x^3 + x^2 + x + 1] [ x + 1 0] sage: key <BLANKLINE> [ x^3 + x^2 x^3 + x^2 + x + 1] [ x + 1 0] sage: maes.round_key(key, 1) <BLANKLINE> [ x + 1 x^3 + x^2 + x + 1] [ 0 x^3 + x^2 + x + 1] sage: maes.round_key(key, 2) <BLANKLINE> [x^2 + x x^3 + 1] [x^2 + x x^2 + x]
TESTS:
Only two rounds are defined for this AES variant::
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, -1) Traceback (most recent call last): ... ValueError: Mini-AES only defines two rounds sage: maes.round_key(key, 3) Traceback (most recent call last): ... ValueError: Mini-AES only defines two rounds
The input key must be a matrix::
sage: maes.round_key("key", 0) Traceback (most recent call last): ... TypeError: secret key must be a 2 x 2 matrix over GF(16)
In addition, the dimensions of the key matrix must be `2 \times 2`::
sage: K = FiniteField(16, "x") sage: MS = MatrixSpace(K, 1, 2) sage: key = MS([[K("x^3 + x^2 + x + 1"), K("0")]]) sage: maes.round_key(key, 2) Traceback (most recent call last): ... TypeError: secret key must be a 2 x 2 matrix over GF(16) """ not (key.base_ring().order() == 16 and key.base_ring().is_field()):
# round 0 # round 1 # round 2 # unsupported round number
r""" 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) """
r""" 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
.. MATH::
\begin{bmatrix} b_0 & b_2 \\ b_1 & b_3 \end{bmatrix}
Then the operation of shift-row is the mapping
.. MATH::
\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}
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) <BLANKLINE> [x^3 + x^2 + x + 1 0] [ x^3 + x x^2 + x + 1] sage: mat <BLANKLINE> [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 <BLANKLINE> [ x^2 + 1 1] [x^2 + x + 1 x^2] sage: maes.shift_row(B) <BLANKLINE> [ 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 <BLANKLINE> [ x + 1 x^2 + x] [ x^3 + 1 x^3 + x^2] sage: maes.shift_row(P) <BLANKLINE> [ x + 1 x^2 + x] [x^3 + x^2 x^3 + 1]
TESTS:
The input block must be a matrix::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes.shift_row("block") Traceback (most recent call last): ... TypeError: input block must be a 2 x 2 matrix over GF(16)
In addition, the dimensions of the input matrix must be `2 \times 2`::
sage: K = FiniteField(16, "x") sage: MS = MatrixSpace(K, 1, 2) sage: mat = MS([[K("x^3 + x^2 + x + 1"), K("0")]]) sage: maes.shift_row(mat) Traceback (most recent call last): ... TypeError: input block must be a 2 x 2 matrix over GF(16) """ not (block.base_ring().order() == 16 and block.base_ring().is_field()):
[block[1][1], block[1][0]] ] )
### conversion functions to convert between different data formats
r""" 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:
.. MATH::
\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}
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))] <BLANKLINE> [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 <BLANKLINE> [ 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 <BLANKLINE> [ 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
TESTS:
Input must be an element of `\GF{2^4}`::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: K = FiniteField(8, "x") sage: G = K.random_element() sage: maes.GF_to_binary(G) Traceback (most recent call last): ... TypeError: input G must be an element of GF(16), a list of elements of GF(16), or a matrix over GF(16)
A list of elements belonging to `\GF{2^4}`::
sage: maes.GF_to_binary([]) Traceback (most recent call last): ... ValueError: input G must be an element of GF(16), a list of elements of GF(16), or a matrix over GF(16) sage: G = [K.random_element() for i in range(5)] sage: maes.GF_to_binary(G) Traceback (most recent call last): ... KeyError:...
A matrix over `\GF{2^4}`::
sage: MS = MatrixSpace(FiniteField(7, "x"), 4, 5) sage: maes.GF_to_binary(MS.random_element()) Traceback (most recent call last): ... TypeError: input G must be an element of GF(16), a list of elements of GF(16), or a matrix over GF(16) """ # G is an element over GF(16) # G is a list of elements over GF(16) # G is a matrix over GF(16) # the type of G doesn't match the supported types else:
r""" 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:
.. MATH::
\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}
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 <BLANKLINE> [ 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 <BLANKLINE> [ 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]
TESTS:
Input must be an element of `\GF{2^4}`::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: K = FiniteField(7, "x") sage: G = K.random_element() sage: maes.GF_to_integer(G) Traceback (most recent call last): ... TypeError: input G must be an element of GF(16), a list of elements of GF(16), or a matrix over GF(16)
A list of elements belonging to `\GF{2^4}`::
sage: maes.GF_to_integer([]) Traceback (most recent call last): ... ValueError: input G must be an element of GF(16), a list of elements of GF(16), or a matrix over GF(16) sage: G = [K.random_element() for i in range(5)] sage: maes.GF_to_integer(G) Traceback (most recent call last): ... KeyError:...
A matrix over `\GF{2^4}`::
sage: MS = MatrixSpace(FiniteField(7, "x"), 4, 5) sage: maes.GF_to_integer(MS.random_element()) Traceback (most recent call last): ... TypeError: input G must be an element of GF(16), a list of elements of GF(16), or a matrix over GF(16) """ # G is an element over GF(16) # G is a list of elements over GF(16) # G is a matrix over GF(16) # the type of G doesn't match the supported types else:
r""" 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:
.. MATH::
\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}
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) <BLANKLINE> [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]
TESTS:
The input ``B`` must be a non-empty binary string, where the number of bits is a multiple of 4::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes.binary_to_GF("") Traceback (most recent call last): ... ValueError: the number of bits in the binary string B must be positive and a multiple of 4 sage: maes.binary_to_GF("101") Traceback (most recent call last): ... ValueError: the number of bits in the binary string B must be positive and a multiple of 4 """ # an empty string # a string with number of bits that is a multiple of 4 else:
r""" 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:
.. MATH::
\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}
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]
TESTS:
The input ``B`` must be a non-empty binary string, where the number of bits is a multiple of 4::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes.binary_to_integer("") Traceback (most recent call last): ... ValueError: the number of bits in the binary string B must be positive and a multiple of 4 sage: maes.binary_to_integer("101") Traceback (most recent call last): ... ValueError: the number of bits in the binary string B must be positive and a multiple of 4 """ # an empty string # a string with number of bits that is a multiple of 4 else:
r""" 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:
.. MATH::
\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}
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
TESTS:
The input ``N`` can be an integer, but must be bounded such that `0 \leq N \leq 15`::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes.integer_to_binary(-1) Traceback (most recent call last): ... KeyError:... sage: maes.integer_to_binary("1") Traceback (most recent call last): ... TypeError: N must be an integer 0 <= N <= 15 or a list of such integers sage: maes.integer_to_binary("") Traceback (most recent call last): ... TypeError: N must be an integer 0 <= N <= 15 or a list of such integers
The input ``N`` can be a list of integers, but each integer `n` of the list must be `0 \leq n \leq 15`::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes.integer_to_binary([]) Traceback (most recent call last): ... ValueError: N must be an integer 0 <= N <= 15 or a list of such integers sage: maes.integer_to_binary([""]) Traceback (most recent call last): ... KeyError:... sage: maes.integer_to_binary([0, 1, 2, 16]) Traceback (most recent call last): ... KeyError:... """ # Here, we assume that each element of the list is an integer n # such that 0 <= n <= 15. An error will be raised if otherwise. # Here, we assume that N is an integer such that 0 <= n <= 15. # An error will be raised if otherwise. else:
r""" 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:
.. MATH::
\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}
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) <BLANKLINE> [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]
TESTS:
The input ``N`` can be an integer, but it must be such that `0 \leq N \leq 15`::
sage: from sage.crypto.block_cipher.miniaes import MiniAES sage: maes = MiniAES() sage: maes.integer_to_GF(-1) Traceback (most recent call last): ... KeyError:... sage: maes.integer_to_GF(16) Traceback (most recent call last): ... KeyError:... sage: maes.integer_to_GF("2") Traceback (most recent call last): ... TypeError: N must be an integer 0 <= N <= 15 or a list of such integers
The input ``N`` can be a list of integers, but each integer `n` in the list must be bounded such that `0 \leq n \leq 15`::
sage: maes.integer_to_GF([]) Traceback (most recent call last): ... ValueError: N must be an integer 0 <= N <= 15 or a list of such integers sage: maes.integer_to_GF([""]) Traceback (most recent call last): ... KeyError:... sage: maes.integer_to_GF([0, 2, 3, "4"]) Traceback (most recent call last): ... KeyError:... sage: maes.integer_to_GF([0, 2, 3, 16]) Traceback (most recent call last): ... KeyError:... """ # Here, we assume that each element of the list is an integer n # such that 0 <= n <= 15. An error will be raised if otherwise. # Here, we assume that N is an integer such that 0 <= n <= 15. # An error will be raised if otherwise. else: |