Rijndael MixColumns

Short description: Cryptographic operation in the Rijndael encryption algorithm

The MixColumns operation performed by the Rijndael cipher, along with the ShiftRows step, is the primary source of diffusion in Rijndael. Each column is treated as a four-term polynomial $b (x) = b_{3} x^{3} + b_{2} x^{2} + b_{1} x + b_{0}$ which are elements within the field $GF (2^{8})$ . The coefficients of the polynomials are elements within the prime sub-field $GF (2)$ .

Each column is multiplied with a fixed polynomial $a (x) = 3 x^{3} + x^{2} + x + 2$ modulo $x^{4} + 1$ ; the inverse of this polynomial is $a^{- 1} (x) = 11 x^{3} + 13 x^{2} + 9 x + 14$ .

Demonstration

The polynomial $a (x) = 3 x^{3} + x^{2} + x + 2$ will be expressed as $a (x) = a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0}$ .

Polynomial multiplication

\begin{aligned} a (x) ∙ b (x) = c (x) & = (a_{3} x^{3} + a_{2} x^{2} + a_{1} x + a_{0}) ∙ (b_{3} x^{3} + b_{2} x^{2} + b_{1} x + b_{0}) \\ = c_{6} x^{6} + c_{5} x^{5} + c_{4} x^{4} + c_{3} x^{3} + c_{2} x^{2} + c_{1} x + c_{0} \end{aligned}

where:

\begin{aligned} c_{0} & = a_{0} ∙ b_{0} \\ c_{1} & = a_{1} ∙ b_{0} \oplus a_{0} ∙ b_{1} \\ c_{2} & = a_{2} ∙ b_{0} \oplus a_{1} ∙ b_{1} \oplus a_{0} ∙ b_{2} \\ c_{3} & = a_{3} ∙ b_{0} \oplus a_{2} ∙ b_{1} \oplus a_{1} ∙ b_{2} \oplus a_{0} ∙ b_{3} \\ c_{4} & = a_{3} ∙ b_{1} \oplus a_{2} ∙ b_{2} \oplus a_{1} ∙ b_{3} \\ c_{5} & = a_{3} ∙ b_{2} \oplus a_{2} ∙ b_{3} \\ c_{6} & = a_{3} ∙ b_{3} \end{aligned}

Modular reduction

The result $c (x)$ is a seven-term polynomial, which must be reduced to a four-byte word, which is done by doing the multiplication modulo $x^{4} + 1$ .

If we do some basic polynomial modular operations we can see that:

\begin{aligned} x^{6} mod (x^{4} + 1) & = - x^{2} = x^{2} over GF (2^{8}) \\ x^{5} mod (x^{4} + 1) & = - x = x over GF (2^{8}) \\ x^{4} mod (x^{4} + 1) & = - 1 = 1 over GF (2^{8}) \end{aligned}

In general, we can say that $x^{i} mod (x^{4} + 1) = x^{i mod 4} .$

So

\begin{aligned} a (x) \otimes b (x) & = c (x) mod (x^{4} + 1) \\ = (c_{6} x^{6} + c_{5} x^{5} + c_{4} x^{4} + c_{3} x^{3} + c_{2} x^{2} + c_{1} x + c_{0}) mod (x^{4} + 1) \\ = c_{6} x^{6 mod 4} + c_{5} x^{5 mod 4} + c_{4} x^{4 mod 4} + c_{3} x^{3 mod 4} + c_{2} x^{2 mod 4} + c_{1} x^{1 mod 4} + c_{0} x^{0 mod 4} \\ = c_{6} x^{2} + c_{5} x + c_{4} + c_{3} x^{3} + c_{2} x^{2} + c_{1} x + c_{0} \\ = c_{3} x^{3} + (c_{2} \oplus c_{6}) x^{2} + (c_{1} \oplus c_{5}) x + c_{0} \oplus c_{4} \\ = d_{3} x^{3} + d_{2} x^{2} + d_{1} x + d_{0} \end{aligned}

where

d_{0} = c_{0} \oplus c_{4}

d_{1} = c_{1} \oplus c_{5}

d_{2} = c_{2} \oplus c_{6}

d_{3} = c_{3}

Matrix representation

The coefficient $d_{3}$ , $d_{2}$ , $d_{1}$ and $d_{0}$ can also be expressed as follows:

d_{0} = a_{0} ∙ b_{0} \oplus a_{3} ∙ b_{1} \oplus a_{2} ∙ b_{2} \oplus a_{1} ∙ b_{3}

d_{1} = a_{1} ∙ b_{0} \oplus a_{0} ∙ b_{1} \oplus a_{3} ∙ b_{2} \oplus a_{2} ∙ b_{3}

d_{2} = a_{2} ∙ b_{0} \oplus a_{1} ∙ b_{1} \oplus a_{0} ∙ b_{2} \oplus a_{3} ∙ b_{3}

d_{3} = a_{3} ∙ b_{0} \oplus a_{2} ∙ b_{1} \oplus a_{1} ∙ b_{2} \oplus a_{0} ∙ b_{3}

And when we replace the coefficients of $a (x)$ with the constants $[\begin{matrix} 3 & 1 & 1 & 2 \end{matrix}]$ used in the cipher we obtain the following:

d_{0} = 2 ∙ b_{0} \oplus 3 ∙ b_{1} \oplus 1 ∙ b_{2} \oplus 1 ∙ b_{3}

d_{1} = 1 ∙ b_{0} \oplus 2 ∙ b_{1} \oplus 3 ∙ b_{2} \oplus 1 ∙ b_{3}

d_{2} = 1 ∙ b_{0} \oplus 1 ∙ b_{1} \oplus 2 ∙ b_{2} \oplus 3 ∙ b_{3}

d_{3} = 3 ∙ b_{0} \oplus 1 ∙ b_{1} \oplus 1 ∙ b_{2} \oplus 2 ∙ b_{3}

This demonstrates that the operation itself is similar to a Hill cipher. It can be performed by multiplying a coordinate vector of four numbers in Rijndael's Galois field by the following circulant MDS matrix:

[\begin{matrix} d_{0} \\ d_{1} \\ d_{2} \\ d_{3} \end{matrix}] = [\begin{matrix} 2 & 3 & 1 & 1 \\ 1 & 2 & 3 & 1 \\ 1 & 1 & 2 & 3 \\ 3 & 1 & 1 & 2 \end{matrix}] [\begin{matrix} b_{0} \\ b_{1} \\ b_{2} \\ b_{3} \end{matrix}]

Implementation example

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 example of such an implementation follows:

void gmix_column(unsigned char *r) {
    unsigned char a[4];
    unsigned char b[4];
    unsigned char c;
    unsigned char h;
    /* The array 'a' is simply a copy of the input array 'r'
     * The array 'b' is each element of the array 'a' multiplied by 2
     * in Rijndael's Galois field
     * a[n] ^ b[n] is element n multiplied by 3 in Rijndael's Galois field */ 
    for (c = 0; c < 4; c++) {
        a[c] = r[c];
        /* h is set to 0x01 if the high bit of r[c] is set, 0x00 otherwise */
        h = r[c] >> 7;    /* logical right shift, thus shifting in zeros */
        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 */
        b[c] ^= h * 0x1B; /* Rijndael's Galois field */
    }
    r[0] = b[0] ^ a[3] ^ a[2] ^ b[1] ^ a[1]; /* 2 * a0 + a3 + a2 + 3 * a1 */
    r[1] = b[1] ^ a[0] ^ a[3] ^ b[2] ^ a[2]; /* 2 * a1 + a0 + a3 + 3 * a2 */
    r[2] = b[2] ^ a[1] ^ a[0] ^ b[3] ^ a[3]; /* 2 * a2 + a1 + a0 + 3 * a3 */
    r[3] = b[3] ^ a[2] ^ a[1] ^ b[0] ^ a[0]; /* 2 * a3 + a2 + a1 + 3 * a0 */
}

A C# example

private byte GMul(byte a, byte b) { // Galois Field (256) Multiplication of two Bytes
    byte p = 0;

    for (int counter = 0; counter < 8; counter++) {
        if ((b & 1) != 0) {
            p ^= a;
        }

        bool hi_bit_set = (a & 0x80) != 0;
        a <<= 1;
        if (hi_bit_set) {
            a ^= 0x1B; /* x^8 + x^4 + x^3 + x + 1 */
        }
        b >>= 1;
    }

    return p;
}

private void MixColumns() { // 's' is the main State matrix, 'ss' is a temp matrix of the same dimensions as 's'.
    Array.Clear(ss, 0, ss.Length);

    for (int c = 0; c < 4; c++) {
        ss[0, c] = (byte)(GMul(0x02, s[0, c]) ^ GMul(0x03, s[1, c]) ^ s[2, c] ^ s[3, c]);
        ss[1, c] = (byte)(s[0, c] ^ GMul(0x02, s[1, c]) ^ GMul(0x03, s[2, c]) ^ s[3,c]);
        ss[2, c] = (byte)(s[0, c] ^ s[1, c] ^ GMul(0x02, s[2, c]) ^ GMul(0x03, s[3, c]));
        ss[3, c] = (byte)(GMul(0x03, s[0,c]) ^ s[1, c] ^ s[2, c] ^ GMul(0x02, s[3, c]));
    }

    ss.CopyTo(s, 0);
}

Test vectors for MixColumn()

Hexadecimal		Decimal
Before	After	Before	After
`db 13 53 45`	`8e 4d a1 bc`	219 19 83 69	142 77 161 188
`f2 0a 22 5c`	`9f dc 58 9d`	242 10 34 92	159 220 88 157
`01 01 01 01`	`01 01 01 01`	1 1 1 1	1 1 1 1
`c6 c6 c6 c6`	`c6 c6 c6 c6`	198 198 198 198	198 198 198 198
`d4 d4 d4 d5`	`d5 d5 d7 d6`	212 212 212 213	213 213 215 214
`2d 26 31 4c`	`4d 7e bd f8`	45 38 49 76	77 126 189 248

InverseMixColumns

The MixColumns operation has the following inverse (numbers are decimal):

[\begin{matrix} b_{0} \\ b_{1} \\ b_{2} \\ b_{3} \end{matrix}] = [\begin{matrix} 14 & 11 & 13 & 9 \\ 9 & 14 & 11 & 13 \\ 13 & 9 & 14 & 11 \\ 11 & 13 & 9 & 14 \end{matrix}] [\begin{matrix} d_{0} \\ d_{1} \\ d_{2} \\ d_{3} \end{matrix}]

Or:

\begin{aligned} b_{0} & = 14 ∙ d_{0} \oplus 11 ∙ d_{1} \oplus 13 ∙ d_{2} \oplus 9 ∙ d_{3} \\ b_{1} & = 9 ∙ d_{0} \oplus 14 ∙ d_{1} \oplus 11 ∙ d_{2} \oplus 13 ∙ d_{3} \\ b_{2} & = 13 ∙ d_{0} \oplus 9 ∙ d_{1} \oplus 14 ∙ d_{2} \oplus 11 ∙ d_{3} \\ b_{3} & = 11 ∙ d_{0} \oplus 13 ∙ d_{1} \oplus 9 ∙ d_{2} \oplus 14 ∙ d_{3} \end{aligned}

Galois Multiplication lookup tables

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.

For instance, in C# these tables can be stored in Byte[256] arrays. In order to compute

p * 3

The result is obtained this way:

result = table_3[(int)p]

Some of the most common instances of these lookup tables are as follows:

Multiply by 2:

0x00,0x02,0x04,0x06,0x08,0x0a,0x0c,0x0e,0x10,0x12,0x14,0x16,0x18,0x1a,0x1c,0x1e,
0x20,0x22,0x24,0x26,0x28,0x2a,0x2c,0x2e,0x30,0x32,0x34,0x36,0x38,0x3a,0x3c,0x3e,
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,	
0x80,0x82,0x84,0x86,0x88,0x8a,0x8c,0x8e,0x90,0x92,0x94,0x96,0x98,0x9a,0x9c,0x9e,
0xa0,0xa2,0xa4,0xa6,0xa8,0xaa,0xac,0xae,0xb0,0xb2,0xb4,0xb6,0xb8,0xba,0xbc,0xbe,
0xc0,0xc2,0xc4,0xc6,0xc8,0xca,0xcc,0xce,0xd0,0xd2,0xd4,0xd6,0xd8,0xda,0xdc,0xde,
0xe0,0xe2,0xe4,0xe6,0xe8,0xea,0xec,0xee,0xf0,0xf2,0xf4,0xf6,0xf8,0xfa,0xfc,0xfe,
0x1b,0x19,0x1f,0x1d,0x13,0x11,0x17,0x15,0x0b,0x09,0x0f,0x0d,0x03,0x01,0x07,0x05,
0x3b,0x39,0x3f,0x3d,0x33,0x31,0x37,0x35,0x2b,0x29,0x2f,0x2d,0x23,0x21,0x27,0x25,
0x5b,0x59,0x5f,0x5d,0x53,0x51,0x57,0x55,0x4b,0x49,0x4f,0x4d,0x43,0x41,0x47,0x45,
0x7b,0x79,0x7f,0x7d,0x73,0x71,0x77,0x75,0x6b,0x69,0x6f,0x6d,0x63,0x61,0x67,0x65,
0x9b,0x99,0x9f,0x9d,0x93,0x91,0x97,0x95,0x8b,0x89,0x8f,0x8d,0x83,0x81,0x87,0x85,
0xbb,0xb9,0xbf,0xbd,0xb3,0xb1,0xb7,0xb5,0xab,0xa9,0xaf,0xad,0xa3,0xa1,0xa7,0xa5,
0xdb,0xd9,0xdf,0xdd,0xd3,0xd1,0xd7,0xd5,0xcb,0xc9,0xcf,0xcd,0xc3,0xc1,0xc7,0xc5,
0xfb,0xf9,0xff,0xfd,0xf3,0xf1,0xf7,0xf5,0xeb,0xe9,0xef,0xed,0xe3,0xe1,0xe7,0xe5

Multiply by 3:

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,
0xc0,0xc3,0xc6,0xc5,0xcc,0xcf,0xca,0xc9,0xd8,0xdb,0xde,0xdd,0xd4,0xd7,0xd2,0xd1,
0xf0,0xf3,0xf6,0xf5,0xfc,0xff,0xfa,0xf9,0xe8,0xeb,0xee,0xed,0xe4,0xe7,0xe2,0xe1,
0xa0,0xa3,0xa6,0xa5,0xac,0xaf,0xaa,0xa9,0xb8,0xbb,0xbe,0xbd,0xb4,0xb7,0xb2,0xb1,
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,	
0xcb,0xc8,0xcd,0xce,0xc7,0xc4,0xc1,0xc2,0xd3,0xd0,0xd5,0xd6,0xdf,0xdc,0xd9,0xda,	
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

Multiply by 9:

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,
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,
0x76,0x7f,0x64,0x6d,0x52,0x5b,0x40,0x49,0x3e,0x37,0x2c,0x25,0x1a,0x13,0x08,0x01,
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

Multiply by 11 (0xB):

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

Multiply by 13 (0xD):

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

Multiply by 14 (0xE):

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

References

FIPS PUB 197: the official AES standard (PDF file)

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Rijndael MixColumns

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Demonstration

Polynomial multiplication

Modular reduction

Matrix representation

Implementation example

Test vectors for MixColumn()

InverseMixColumns

Galois Multiplication lookup tables

References

See also

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Rijndael MixColumns

Demonstration

Polynomial multiplication

Modular reduction

Matrix representation

Implementation example

Test vectors for MixColumn()

InverseMixColumns

Galois Multiplication lookup tables

References

See also

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