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| The MixColumns operation performed by the [[Rijndael]] cipher, along with the shift-rows step, is the primary source of [[confusion and diffusion|diffusion]] in Rijndael. Each column is treated as a polynomial over '''GF'''(''2<sup>8</sup>'') and is then multiplied modulo <math>x^4+1</math> with a fixed polynomial <math>c(x) = 3x^3 + x^2 + x + 2</math>; the inverse of this polynomial is <math>c^{-1}(x) = 11x^3 + 13x^2 + 9x + 14</math>.
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| ==MixColumns==
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| The MixColumns step can be performed by multiplying a [[coordinate vector]] of four numbers in [[Rijndael Galois field|Rijndael's Galois field]] by the following [[Circulant_matrix|circulant]] [[MDS matrix]]:
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| :<math>\begin{bmatrix}b_0\\b_1\\b_2\\b_3\end{bmatrix} =
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| \begin{bmatrix}
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| 2&3&1&1 \\
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| 1&2&3&1 \\
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| 1&1&2&3 \\
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| 3&1&1&2 \end{bmatrix} \begin{bmatrix}a_0\\a_1\\a_2\\a_3\end{bmatrix}</math>
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| This can also be seen as the following:
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| :<math>b_0 = 2a_0 + 3a_1 + 1a_2 + 1a_3</math>
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| :<math>b_1 = 1a_0 + 2a_1 + 3a_2 + 1a_3</math>
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| :<math>b_2 = 1a_0 + 1a_1 + 2a_2 + 3a_3</math>
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| :<math>b_3 = 3a_0 + 1a_1 + 1a_2 + 2a_3</math>
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| Since this math is done in [[Rijndael Galois field|Rijndael's Galois field]], the addition above is actually an [[exclusive or]] operation, and multiplication is a [[Finite field arithmetic#Multiplication|complicated operation]].
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| ==Implementation example==
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| This can be simplified somewhat in actual implementation by replacing the multiply by 2 with a single shift and conditional exclusive or, and replacing a multiply by 3 with a multiply by 2 combined with an exclusive or. A [[C (programming language)|C]] example of such an implementation follows:
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| <pre>
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| void gmix_column(unsigned char *r) {
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| unsigned char a[4];
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| unsigned char b[4];
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| unsigned char c;
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| unsigned char h;
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| /* The array 'a' is simply a copy of the input array 'r'
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| * The array 'b' is each element of the array 'a' multiplied by 2
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| * in Rijndael's Galois field
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| * a[n] ^ b[n] is element n multiplied by 3 in Rijndael's Galois field */
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| for(c=0;c<4;c++) {
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| a[c] = r[c];
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| /* h is 0xff if the high bit of r[c] is set, 0 otherwise */
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| h = (unsigned char)((signed char)r[c] >> 7); /* arithmetic right shift, thus shifting in either zeros or ones */
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| b[c] = r[c] << 1; /* implicitly removes high bit because b[c] is an 8-bit char, so we xor by 0x1b and not 0x11b in the next line */
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| b[c] ^= 0x1B & h; /* Rijndael's Galois field */
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| }
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| r[0] = b[0] ^ a[3] ^ a[2] ^ b[1] ^ a[1]; /* 2 * a0 + a3 + a2 + 3 * a1 */
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| r[1] = b[1] ^ a[0] ^ a[3] ^ b[2] ^ a[2]; /* 2 * a1 + a0 + a3 + 3 * a2 */
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| r[2] = b[2] ^ a[1] ^ a[0] ^ b[3] ^ a[3]; /* 2 * a2 + a1 + a0 + 3 * a3 */
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| r[3] = b[3] ^ a[2] ^ a[1] ^ b[0] ^ a[0]; /* 2 * a3 + a2 + a1 + 3 * a0 */
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| }
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| </pre>
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| A C# example
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| <pre> | |
| private Byte GMul(Byte a, Byte b) { // Galois Field (256) Multiplication of two Bytes
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| Byte p = 0;
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| Byte counter;
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| Byte hi_bit_set;
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| for (counter = 0; counter < 8; counter++) {
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| if ((b & 1) != 0) {
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| p ^= a;
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| }
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| hi_bit_set = (Byte) (a & 0x80);
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| a <<= 1;
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| if (hi_bit_set != 0) {
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| a ^= 0x1b; /* x^8 + x^4 + x^3 + x + 1 */
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| }
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| b >>= 1;
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| }
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| return p;
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| }
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| private void MixColumns() { // 's' is the main State matrix, 'ss' is a temp matrix of the same dimensions as 's'.
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| Array.Clear(ss, 0, ss.Length);
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| for (int c = 0; c < 4; c++) {
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| ss[0, c] = (Byte) (GMul(0x02, s[0, c]) ^ GMul(0x03, s[1, c]) ^ s[2, c] ^ s[3, c]);
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| ss[1, c] = (Byte) (s[0, c] ^ GMul(0x02, s[1, c]) ^ GMul(0x03, s[2, c]) ^ s[3,c]);
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| ss[2, c] = (Byte) (s[0, c] ^ s[1, c] ^ GMul(0x02, s[2, c]) ^ GMul(0x03, s[3, c]));
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| ss[3, c] = (Byte) (GMul(0x03, s[0,c]) ^ s[1, c] ^ s[2, c] ^ GMul(0x02, s[3, c]));
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| }
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| ss.CopyTo(s, 0);
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| }
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| </pre>
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| ==InverseMixColumns==
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| The MixColumns operation has the following inverse (numbers are decimal):
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| :<math>\begin{bmatrix}r_0\\r_1\\r_2\\r_3\end{bmatrix} = \begin{bmatrix}
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| 14&11&13&9 \\
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| 9&14&11&13 \\
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| 13&9&14&11 \\
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| 11&13&9&14 \end{bmatrix} \begin{bmatrix}a_0\\a_1\\a_2\\a_3\end{bmatrix}</math>
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| Or:
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| :<math>r_0 = 14a_0 + 11a_1 + 13a_2 + 9a_3</math>
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| :<math>r_1 = 9a_0 + 14a_1 + 11a_2 + 13a_3</math>
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| :<math>r_2 = 13a_0 + 9a_1 + 14a_2 + 11a_3</math>
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| :<math>r_3 = 11a_0 + 13a_1 + 9a_2 + 14a_3</math>
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| ==Test vectors for MixColumns(); not for InvMixColumns ==
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| {|class="wikitable"
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| !colspan=2|Hexadecimal
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| !colspan=2|Decimal
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| |-
| |
| !Before
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| !After
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| !Before
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| !After
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| |-
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| |<tt>db 13 53 45</tt>
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| |<tt>8e 4d a1 bc</tt>
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| |219 19 83 69
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| |142 77 161 188
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| |-
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| |<tt>f2 0a 22 5c</tt>
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| |<tt>9f dc 58 9d</tt>
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| |242 10 34 92
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| |159 220 88 157
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| |-
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| |<tt>01 01 01 01</tt>
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| |<tt>01 01 01 01</tt>
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| |1 1 1 1
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| |1 1 1 1
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| |-
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| |<tt>c6 c6 c6 c6</tt>
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| |<tt>c6 c6 c6 c6</tt>
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| |198 198 198 198
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| |198 198 198 198
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| |-
| |
| |<tt>d4 d4 d4 d5</tt>
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| |<tt>d5 d5 d7 d6</tt>
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| |212 212 212 213
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| |213 213 215 214
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| |-
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| |<tt>2d 26 31 4c</tt>
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| |<tt>4d 7e bd f8</tt>
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| |45 38 49 76
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| |77 126 189 248
| |
| |}
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| == Galois Multiplication lookup tables ==
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| Commonly, rather than implementing galois multiplication, Rijndael implementations simply use pre-calculated lookup tables to perform the byte multiplication by 2, 3, 9, 11, 13, and 14.
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| For instance, in C# these tables can be stored in Byte[256] arrays. In order to compute
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| <pre>p * 3</pre>
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| The result is obtained this way:
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| <pre>result = table_3[(int)p]</pre>
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| These lookup tables are as follows:
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| Multiply by 2:
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| <pre>0x00,0x02,0x04,0x06,0x08,0x0a,0x0c,0x0e,0x10,0x12,0x14,0x16,0x18,0x1a,0x1c,0x1e,
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| 0x20,0x22,0x24,0x26,0x28,0x2a,0x2c,0x2e,0x30,0x32,0x34,0x36,0x38,0x3a,0x3c,0x3e,
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| 0x40,0x42,0x44,0x46,0x48,0x4a,0x4c,0x4e,0x50,0x52,0x54,0x56,0x58,0x5a,0x5c,0x5e,
| |
| 0x60,0x62,0x64,0x66,0x68,0x6a,0x6c,0x6e,0x70,0x72,0x74,0x76,0x78,0x7a,0x7c,0x7e,
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| 0x80,0x82,0x84,0x86,0x88,0x8a,0x8c,0x8e,0x90,0x92,0x94,0x96,0x98,0x9a,0x9c,0x9e,
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| 0xa0,0xa2,0xa4,0xa6,0xa8,0xaa,0xac,0xae,0xb0,0xb2,0xb4,0xb6,0xb8,0xba,0xbc,0xbe,
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| 0xc0,0xc2,0xc4,0xc6,0xc8,0xca,0xcc,0xce,0xd0,0xd2,0xd4,0xd6,0xd8,0xda,0xdc,0xde,
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| 0xe0,0xe2,0xe4,0xe6,0xe8,0xea,0xec,0xee,0xf0,0xf2,0xf4,0xf6,0xf8,0xfa,0xfc,0xfe,
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| 0x1b,0x19,0x1f,0x1d,0x13,0x11,0x17,0x15,0x0b,0x09,0x0f,0x0d,0x03,0x01,0x07,0x05,
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| 0x3b,0x39,0x3f,0x3d,0x33,0x31,0x37,0x35,0x2b,0x29,0x2f,0x2d,0x23,0x21,0x27,0x25,
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| 0x5b,0x59,0x5f,0x5d,0x53,0x51,0x57,0x55,0x4b,0x49,0x4f,0x4d,0x43,0x41,0x47,0x45,
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| 0x7b,0x79,0x7f,0x7d,0x73,0x71,0x77,0x75,0x6b,0x69,0x6f,0x6d,0x63,0x61,0x67,0x65,
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| 0x9b,0x99,0x9f,0x9d,0x93,0x91,0x97,0x95,0x8b,0x89,0x8f,0x8d,0x83,0x81,0x87,0x85,
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| 0xbb,0xb9,0xbf,0xbd,0xb3,0xb1,0xb7,0xb5,0xab,0xa9,0xaf,0xad,0xa3,0xa1,0xa7,0xa5,
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| 0xdb,0xd9,0xdf,0xdd,0xd3,0xd1,0xd7,0xd5,0xcb,0xc9,0xcf,0xcd,0xc3,0xc1,0xc7,0xc5,
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| 0xfb,0xf9,0xff,0xfd,0xf3,0xf1,0xf7,0xf5,0xeb,0xe9,0xef,0xed,0xe3,0xe1,0xe7,0xe5</pre>
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| Multiply by 3:
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| <pre>0x00,0x03,0x06,0x05,0x0c,0x0f,0x0a,0x09,0x18,0x1b,0x1e,0x1d,0x14,0x17,0x12,0x11,
| |
| 0x30,0x33,0x36,0x35,0x3c,0x3f,0x3a,0x39,0x28,0x2b,0x2e,0x2d,0x24,0x27,0x22,0x21,
| |
| 0x60,0x63,0x66,0x65,0x6c,0x6f,0x6a,0x69,0x78,0x7b,0x7e,0x7d,0x74,0x77,0x72,0x71,
| |
| 0x50,0x53,0x56,0x55,0x5c,0x5f,0x5a,0x59,0x48,0x4b,0x4e,0x4d,0x44,0x47,0x42,0x41,
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| 0xc0,0xc3,0xc6,0xc5,0xcc,0xcf,0xca,0xc9,0xd8,0xdb,0xde,0xdd,0xd4,0xd7,0xd2,0xd1,
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| 0xf0,0xf3,0xf6,0xf5,0xfc,0xff,0xfa,0xf9,0xe8,0xeb,0xee,0xed,0xe4,0xe7,0xe2,0xe1,
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| 0xa0,0xa3,0xa6,0xa5,0xac,0xaf,0xaa,0xa9,0xb8,0xbb,0xbe,0xbd,0xb4,0xb7,0xb2,0xb1,
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| 0x90,0x93,0x96,0x95,0x9c,0x9f,0x9a,0x99,0x88,0x8b,0x8e,0x8d,0x84,0x87,0x82,0x81,
| |
| 0x9b,0x98,0x9d,0x9e,0x97,0x94,0x91,0x92,0x83,0x80,0x85,0x86,0x8f,0x8c,0x89,0x8a,
| |
| 0xab,0xa8,0xad,0xae,0xa7,0xa4,0xa1,0xa2,0xb3,0xb0,0xb5,0xb6,0xbf,0xbc,0xb9,0xba,
| |
| 0xfb,0xf8,0xfd,0xfe,0xf7,0xf4,0xf1,0xf2,0xe3,0xe0,0xe5,0xe6,0xef,0xec,0xe9,0xea,
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| 0xcb,0xc8,0xcd,0xce,0xc7,0xc4,0xc1,0xc2,0xd3,0xd0,0xd5,0xd6,0xdf,0xdc,0xd9,0xda,
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| 0x5b,0x58,0x5d,0x5e,0x57,0x54,0x51,0x52,0x43,0x40,0x45,0x46,0x4f,0x4c,0x49,0x4a,
| |
| 0x6b,0x68,0x6d,0x6e,0x67,0x64,0x61,0x62,0x73,0x70,0x75,0x76,0x7f,0x7c,0x79,0x7a,
| |
| 0x3b,0x38,0x3d,0x3e,0x37,0x34,0x31,0x32,0x23,0x20,0x25,0x26,0x2f,0x2c,0x29,0x2a,
| |
| 0x0b,0x08,0x0d,0x0e,0x07,0x04,0x01,0x02,0x13,0x10,0x15,0x16,0x1f,0x1c,0x19,0x1a</pre>
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| Multiply by 9:
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| <pre>0x00,0x09,0x12,0x1b,0x24,0x2d,0x36,0x3f,0x48,0x41,0x5a,0x53,0x6c,0x65,0x7e,0x77,
| |
| 0x90,0x99,0x82,0x8b,0xb4,0xbd,0xa6,0xaf,0xd8,0xd1,0xca,0xc3,0xfc,0xf5,0xee,0xe7,
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| 0x3b,0x32,0x29,0x20,0x1f,0x16,0x0d,0x04,0x73,0x7a,0x61,0x68,0x57,0x5e,0x45,0x4c,
| |
| 0xab,0xa2,0xb9,0xb0,0x8f,0x86,0x9d,0x94,0xe3,0xea,0xf1,0xf8,0xc7,0xce,0xd5,0xdc,
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| 0x76,0x7f,0x64,0x6d,0x52,0x5b,0x40,0x49,0x3e,0x37,0x2c,0x25,0x1a,0x13,0x08,0x01,
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| 0xe6,0xef,0xf4,0xfd,0xc2,0xcb,0xd0,0xd9,0xae,0xa7,0xbc,0xb5,0x8a,0x83,0x98,0x91,
| |
| 0x4d,0x44,0x5f,0x56,0x69,0x60,0x7b,0x72,0x05,0x0c,0x17,0x1e,0x21,0x28,0x33,0x3a,
| |
| 0xdd,0xd4,0xcf,0xc6,0xf9,0xf0,0xeb,0xe2,0x95,0x9c,0x87,0x8e,0xb1,0xb8,0xa3,0xaa,
| |
| 0xec,0xe5,0xfe,0xf7,0xc8,0xc1,0xda,0xd3,0xa4,0xad,0xb6,0xbf,0x80,0x89,0x92,0x9b,
| |
| 0x7c,0x75,0x6e,0x67,0x58,0x51,0x4a,0x43,0x34,0x3d,0x26,0x2f,0x10,0x19,0x02,0x0b,
| |
| 0xd7,0xde,0xc5,0xcc,0xf3,0xfa,0xe1,0xe8,0x9f,0x96,0x8d,0x84,0xbb,0xb2,0xa9,0xa0,
| |
| 0x47,0x4e,0x55,0x5c,0x63,0x6a,0x71,0x78,0x0f,0x06,0x1d,0x14,0x2b,0x22,0x39,0x30,
| |
| 0x9a,0x93,0x88,0x81,0xbe,0xb7,0xac,0xa5,0xd2,0xdb,0xc0,0xc9,0xf6,0xff,0xe4,0xed,
| |
| 0x0a,0x03,0x18,0x11,0x2e,0x27,0x3c,0x35,0x42,0x4b,0x50,0x59,0x66,0x6f,0x74,0x7d,
| |
| 0xa1,0xa8,0xb3,0xba,0x85,0x8c,0x97,0x9e,0xe9,0xe0,0xfb,0xf2,0xcd,0xc4,0xdf,0xd6,
| |
| 0x31,0x38,0x23,0x2a,0x15,0x1c,0x07,0x0e,0x79,0x70,0x6b,0x62,0x5d,0x54,0x4f,0x46</pre>
| |
| | |
| Multiply by 11:
| |
| <pre>0x00,0x0b,0x16,0x1d,0x2c,0x27,0x3a,0x31,0x58,0x53,0x4e,0x45,0x74,0x7f,0x62,0x69,
| |
| 0xb0,0xbb,0xa6,0xad,0x9c,0x97,0x8a,0x81,0xe8,0xe3,0xfe,0xf5,0xc4,0xcf,0xd2,0xd9,
| |
| 0x7b,0x70,0x6d,0x66,0x57,0x5c,0x41,0x4a,0x23,0x28,0x35,0x3e,0x0f,0x04,0x19,0x12,
| |
| 0xcb,0xc0,0xdd,0xd6,0xe7,0xec,0xf1,0xfa,0x93,0x98,0x85,0x8e,0xbf,0xb4,0xa9,0xa2,
| |
| 0xf6,0xfd,0xe0,0xeb,0xda,0xd1,0xcc,0xc7,0xae,0xa5,0xb8,0xb3,0x82,0x89,0x94,0x9f,
| |
| 0x46,0x4d,0x50,0x5b,0x6a,0x61,0x7c,0x77,0x1e,0x15,0x08,0x03,0x32,0x39,0x24,0x2f,
| |
| 0x8d,0x86,0x9b,0x90,0xa1,0xaa,0xb7,0xbc,0xd5,0xde,0xc3,0xc8,0xf9,0xf2,0xef,0xe4,
| |
| 0x3d,0x36,0x2b,0x20,0x11,0x1a,0x07,0x0c,0x65,0x6e,0x73,0x78,0x49,0x42,0x5f,0x54,
| |
| 0xf7,0xfc,0xe1,0xea,0xdb,0xd0,0xcd,0xc6,0xaf,0xa4,0xb9,0xb2,0x83,0x88,0x95,0x9e,
| |
| 0x47,0x4c,0x51,0x5a,0x6b,0x60,0x7d,0x76,0x1f,0x14,0x09,0x02,0x33,0x38,0x25,0x2e,
| |
| 0x8c,0x87,0x9a,0x91,0xa0,0xab,0xb6,0xbd,0xd4,0xdf,0xc2,0xc9,0xf8,0xf3,0xee,0xe5,
| |
| 0x3c,0x37,0x2a,0x21,0x10,0x1b,0x06,0x0d,0x64,0x6f,0x72,0x79,0x48,0x43,0x5e,0x55,
| |
| 0x01,0x0a,0x17,0x1c,0x2d,0x26,0x3b,0x30,0x59,0x52,0x4f,0x44,0x75,0x7e,0x63,0x68,
| |
| 0xb1,0xba,0xa7,0xac,0x9d,0x96,0x8b,0x80,0xe9,0xe2,0xff,0xf4,0xc5,0xce,0xd3,0xd8,
| |
| 0x7a,0x71,0x6c,0x67,0x56,0x5d,0x40,0x4b,0x22,0x29,0x34,0x3f,0x0e,0x05,0x18,0x13,
| |
| 0xca,0xc1,0xdc,0xd7,0xe6,0xed,0xf0,0xfb,0x92,0x99,0x84,0x8f,0xbe,0xb5,0xa8,0xa3</pre>
| |
| | |
| Multiply by 13:
| |
| <pre>0x00,0x0d,0x1a,0x17,0x34,0x39,0x2e,0x23,0x68,0x65,0x72,0x7f,0x5c,0x51,0x46,0x4b,
| |
| 0xd0,0xdd,0xca,0xc7,0xe4,0xe9,0xfe,0xf3,0xb8,0xb5,0xa2,0xaf,0x8c,0x81,0x96,0x9b,
| |
| 0xbb,0xb6,0xa1,0xac,0x8f,0x82,0x95,0x98,0xd3,0xde,0xc9,0xc4,0xe7,0xea,0xfd,0xf0,
| |
| 0x6b,0x66,0x71,0x7c,0x5f,0x52,0x45,0x48,0x03,0x0e,0x19,0x14,0x37,0x3a,0x2d,0x20,
| |
| 0x6d,0x60,0x77,0x7a,0x59,0x54,0x43,0x4e,0x05,0x08,0x1f,0x12,0x31,0x3c,0x2b,0x26,
| |
| 0xbd,0xb0,0xa7,0xaa,0x89,0x84,0x93,0x9e,0xd5,0xd8,0xcf,0xc2,0xe1,0xec,0xfb,0xf6,
| |
| 0xd6,0xdb,0xcc,0xc1,0xe2,0xef,0xf8,0xf5,0xbe,0xb3,0xa4,0xa9,0x8a,0x87,0x90,0x9d,
| |
| 0x06,0x0b,0x1c,0x11,0x32,0x3f,0x28,0x25,0x6e,0x63,0x74,0x79,0x5a,0x57,0x40,0x4d,
| |
| 0xda,0xd7,0xc0,0xcd,0xee,0xe3,0xf4,0xf9,0xb2,0xbf,0xa8,0xa5,0x86,0x8b,0x9c,0x91,
| |
| 0x0a,0x07,0x10,0x1d,0x3e,0x33,0x24,0x29,0x62,0x6f,0x78,0x75,0x56,0x5b,0x4c,0x41,
| |
| 0x61,0x6c,0x7b,0x76,0x55,0x58,0x4f,0x42,0x09,0x04,0x13,0x1e,0x3d,0x30,0x27,0x2a,
| |
| 0xb1,0xbc,0xab,0xa6,0x85,0x88,0x9f,0x92,0xd9,0xd4,0xc3,0xce,0xed,0xe0,0xf7,0xfa,
| |
| 0xb7,0xba,0xad,0xa0,0x83,0x8e,0x99,0x94,0xdf,0xd2,0xc5,0xc8,0xeb,0xe6,0xf1,0xfc,
| |
| 0x67,0x6a,0x7d,0x70,0x53,0x5e,0x49,0x44,0x0f,0x02,0x15,0x18,0x3b,0x36,0x21,0x2c,
| |
| 0x0c,0x01,0x16,0x1b,0x38,0x35,0x22,0x2f,0x64,0x69,0x7e,0x73,0x50,0x5d,0x4a,0x47,
| |
| 0xdc,0xd1,0xc6,0xcb,0xe8,0xe5,0xf2,0xff,0xb4,0xb9,0xae,0xa3,0x80,0x8d,0x9a,0x97</pre>
| |
| | |
| Multiply by 14:
| |
| <pre>0x00,0x0e,0x1c,0x12,0x38,0x36,0x24,0x2a,0x70,0x7e,0x6c,0x62,0x48,0x46,0x54,0x5a,
| |
| 0xe0,0xee,0xfc,0xf2,0xd8,0xd6,0xc4,0xca,0x90,0x9e,0x8c,0x82,0xa8,0xa6,0xb4,0xba,
| |
| 0xdb,0xd5,0xc7,0xc9,0xe3,0xed,0xff,0xf1,0xab,0xa5,0xb7,0xb9,0x93,0x9d,0x8f,0x81,
| |
| 0x3b,0x35,0x27,0x29,0x03,0x0d,0x1f,0x11,0x4b,0x45,0x57,0x59,0x73,0x7d,0x6f,0x61,
| |
| 0xad,0xa3,0xb1,0xbf,0x95,0x9b,0x89,0x87,0xdd,0xd3,0xc1,0xcf,0xe5,0xeb,0xf9,0xf7,
| |
| 0x4d,0x43,0x51,0x5f,0x75,0x7b,0x69,0x67,0x3d,0x33,0x21,0x2f,0x05,0x0b,0x19,0x17,
| |
| 0x76,0x78,0x6a,0x64,0x4e,0x40,0x52,0x5c,0x06,0x08,0x1a,0x14,0x3e,0x30,0x22,0x2c,
| |
| 0x96,0x98,0x8a,0x84,0xae,0xa0,0xb2,0xbc,0xe6,0xe8,0xfa,0xf4,0xde,0xd0,0xc2,0xcc,
| |
| 0x41,0x4f,0x5d,0x53,0x79,0x77,0x65,0x6b,0x31,0x3f,0x2d,0x23,0x09,0x07,0x15,0x1b,
| |
| 0xa1,0xaf,0xbd,0xb3,0x99,0x97,0x85,0x8b,0xd1,0xdf,0xcd,0xc3,0xe9,0xe7,0xf5,0xfb,
| |
| 0x9a,0x94,0x86,0x88,0xa2,0xac,0xbe,0xb0,0xea,0xe4,0xf6,0xf8,0xd2,0xdc,0xce,0xc0,
| |
| 0x7a,0x74,0x66,0x68,0x42,0x4c,0x5e,0x50,0x0a,0x04,0x16,0x18,0x32,0x3c,0x2e,0x20,
| |
| 0xec,0xe2,0xf0,0xfe,0xd4,0xda,0xc8,0xc6,0x9c,0x92,0x80,0x8e,0xa4,0xaa,0xb8,0xb6,
| |
| 0x0c,0x02,0x10,0x1e,0x34,0x3a,0x28,0x26,0x7c,0x72,0x60,0x6e,0x44,0x4a,0x58,0x56,
| |
| 0x37,0x39,0x2b,0x25,0x0f,0x01,0x13,0x1d,0x47,0x49,0x5b,0x55,0x7f,0x71,0x63,0x6d,
| |
| 0xd7,0xd9,0xcb,0xc5,0xef,0xe1,0xf3,0xfd,0xa7,0xa9,0xbb,0xb5,0x9f,0x91,0x83,0x8d</pre>
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| | |
| == References ==
| |
| | |
| * [http://www.iaik.tu-graz.ac.at/research/krypto/AES/old/%7Erijmen/rijndael/rijndaeldocV2.zip Rijndael specification] ([[ZIP (file format)|Zip]] compressed [[Portable Document Format|PDF]] file).
| |
| * [http://www.csrc.nist.gov/publications/fips/fips197/fips-197.pdf FIPS PUB 197: the official AES standard] ([[Portable Document Format|PDF]] file)
| |
| | |
| ==See also==
| |
| * [[Advanced Encryption Standard]]
| |
| | |
| [[Category:Finite fields]]
| |
| [[Category:Advanced Encryption Standard]]
| |
