Rijndael mix columns

MixColumns

The operation consist in the modular multiplication of two four-term polynomials whose coefficients are element of G F ( 2 8 ) . The modulo used for this operation is x 4 + 1.

The first four-term polynomial coefficients is defined by the state column [ b 3 b 2 b 1 b 0 ] which contains four bytes. Each bytes is a coefficient of the four-term polynomial so that b ( x ) = b 3 x 3 + b 2 x 2 + b 1 x + b 0 .

The second four-term polynomial is a constant polynomial a ( x ) = 3 x 3 + x 2 + x + 2 . Its coefficients are also elements of G F ( 2 8 ) , they are here denoted in hexadecimal notation. Its inverse is a − 1 ( x ) = 11 x 3 + 13 x 2 + 9 x + 14 .

Before doing the demonstration, we need to define some notation :

⊗ = multiplication modulo x 4 + 1 ⊕ = addition over G F ( 2 8 ) • = multiplication (usual polynomial multiplication when between polynomials and multiplication over G F ( 2 8 ) for the coefficients)

It is important also to remember that the addition of two polynomials whose coefficients are elements of G F ( 2 8 ) has the following rule :

( 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 ) = ( a 3 ⊕ b 3 ) x 3 + ( a 2 ⊕ b 2 ) x 2 + ( a 1 ⊕ b 1 ) x + ( a 0 ⊕ b 0 )

Demonstration

For the 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 .

STEP 1 : Polynomial multiplication

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

where :

c 0 = a 0 • b 0 c 1 = a 1 • b 0 ⊕ a 0 • b 1 c 2 = a 2 • b 0 ⊕ a 1 • b 1 ⊕ a 0 • b 2 c 3 = a 3 • b 0 ⊕ a 2 • b 1 ⊕ a 1 • b 2 ⊕ a 0 • b 3 c 4 = a 3 • b 1 ⊕ a 2 • b 2 ⊕ a 1 • b 3 c 5 = a 3 • b 2 ⊕ a 2 • b 3 c 6 = a 3 • b 3

STEP 2 : modular reduction

We can see that the result c ( x ) is a seven term polynomial and that it represent a seven-byte word. It needs to be reduced back to a four-byte word, and this is done by doing the multiplication modulo x 4 + 1. .

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

x 6 m o d ( x 4 + 1 ) = − x 2 = x 2 over G F ( 2 8 ) x 5 m o d ( x 4 + 1 ) = − x = x over G F ( 2 8 ) x 4 m o d ( x 4 + 1 ) = − 1 = 1 over G F ( 2 8 )

and from this we can deduce that x i m o d ( x 4 + 1 ) = x ^{i m o d ( 4 )} .

So a ( x ) ⊗ b ( x ) = c ( x ) m o d ( 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 ) m o d ( x 4 + 1 ) = c 6 x ^{6 m o d ( 4 )} + c 5 x ^{5 m o d ( 4 )} + c 4 x ^{4 m o d ( 4 )} + c 3 x ^{3 m o d ( 4 )} + c 2 x ^{2 m o d ( 4 )} + c 1 x ^{1 m o d ( 4 )} + c 0 x ^{0 m o d ( 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 ⊕ c 6 ) x 2 + ( c 1 ⊕ c 5 ) x + c 0 ⊕ c 4 = d 3 x 3 + d 2 x 2 + d 1 x + d 0

where :

d 0 = c 0 ⊕ c 4 d 1 = c 1 ⊕ c 5 d 2 = c 2 ⊕ c 6 d 3 = c 3

STEP 3 : matrix representation

If we use what was done in Step 1, the coefficient d 3 , d 2 , d 1 and d 0 can also be expressed as follow :

d 0 = a 0 • b 0 ⊕ a 3 • b 1 ⊕ a 2 • b 2 ⊕ a 1 • b 3 d 1 = a 1 • b 0 ⊕ a 0 • b 1 ⊕ a 3 • b 2 ⊕ a 2 • b 3 d 2 = a 2 • b 0 ⊕ a 1 • b 1 ⊕ a 0 • b 2 ⊕ a 3 • b 3 d 3 = a 3 • b 0 ⊕ a 2 • b 1 ⊕ a 1 • b 2 ⊕ a 0 • b 3

And when we replace the coefficients of a ( x ) with the constants [ 3 1 1 2 ] used in the cipher we obtain the following :

d 0 = 2 • b 0 ⊕ 3 • b 1 ⊕ 1 • b 2 ⊕ 1 • b 3 d 1 = 1 • b 0 ⊕ 2 • b 1 ⊕ 3 • b 2 ⊕ 1 • b 3 d 2 = 1 • b 0 ⊕ 1 • b 1 ⊕ 2 • b 2 ⊕ 3 • b 3 d 3 = 3 • b 0 ⊕ 1 • b 1 ⊕ 1 • b 2 ⊕ 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:

[ d 0 d 1 d 2 d 3 ] = [ 2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2 ] [ b 0 b 1 b 2 b 3 ]

This can also be seen as the following:

d 0 = 2 b 0 + 3 b 1 + 1 b 2 + 1 b 3 d 1 = 1 b 0 + 2 b 1 + 3 b 2 + 1 b 3 d 2 = 1 b 0 + 1 b 1 + 2 b 2 + 3 b 3 d 3 = 3 b 0 + 1 b 1 + 1 b 2 + 2 b 3

Since this math is done in Rijndael's Galois field, the addition above is actually an exclusive or operation, and multiplication is a complicated operation.

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 0xff if the high bit of r[c] is set, 0 otherwise */ h = (unsigned char)((signed char)r[c] >> 7); /* arithmetic right shift, thus shifting in either zeros or ones */ 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] ^= 0x1B & h; /* 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; Byte counter; Byte hi_bit_set; for (counter = 0; counter < 8; counter++) { if ((b & 1) != 0) { p ^= a; } hi_bit_set = (Byte) (a & 0x80); a <<= 1; if (hi_bit_set != 0) { 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); }

InverseMixColumns

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

[ d 0 d 1 d 2 d 3 ] = [ 14 11 13 9 9 14 11 13 13 9 14 11 11 13 9 14 ] [ b 0 b 1 b 2 b 3 ]

Or:

d 0 = 14 b 0 + 11 b 1 + 13 b 2 + 9 b 3 d 1 = 9 b 0 + 14 b 1 + 11 b 2 + 13 b 3 d 2 = 13 b 0 + 9 b 1 + 14 b 2 + 11 b 3 d 3 = 11 b 0 + 13 b 1 + 9 b 2 + 14 b 3

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]

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:

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:

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:

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