# Rijndael MixColumns

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 ${\displaystyle b(x)=b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}}$, where the coefficients are element over ${\displaystyle \operatorname {GF} (2^{8})}$, and is then multiplied modulo ${\displaystyle x^{4}+1}$ with a fixed polynomial ${\displaystyle a(x)=3x^{3}+x^{2}+x+2}$; the inverse of this polynomial is ${\displaystyle a^{-1}(x)=11x^{3}+13x^{2}+9x+14}$.

## MixColumns

The operation consist in the modular multiplication of two four-term polynomials whose coefficients are element of ${\displaystyle \operatorname {GF} (2^{8})}$. The modulo used for this operation is ${\displaystyle x^{4}+1}$.

The first four-term polynomial coefficients are defined by the state column ${\displaystyle {\begin{bmatrix}b_{3}&b_{2}&b_{1}&b_{0}\end{bmatrix}}}$, which contains four bytes. Each byte is a coefficient of the four-term polynomial so that

${\displaystyle b(x)=b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}.}$

The second four-term polynomial is a constant polynomial ${\displaystyle a(x)=3x^{3}+x^{2}+x+2}$. Its coefficients are also elements of ${\displaystyle \operatorname {GF} (2^{8})}$; they are here denoted in hexadecimal notation. Its inverse is ${\displaystyle a^{-1}(x)=11x^{3}+13x^{2}+9x+14}$.

We need to define some notation:

${\displaystyle \otimes }$ denotes multiplication modulo ${\displaystyle x^{4}+1}$.
${\displaystyle \oplus }$ denotes addition over ${\displaystyle \operatorname {GF} (2^{8})}$.
${\displaystyle \bullet }$ denotes multiplication (usual polynomial multiplication when between polynomials and multiplication over ${\displaystyle \operatorname {GF} (2^{8})}$ for the coefficients).

The addition of two polynomials whose coefficients are elements of ${\displaystyle \operatorname {GF} (2^{8})}$ has the following rule:

{\displaystyle {\begin{aligned}&(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}\oplus b_{3})x^{3}+(a_{2}\oplus b_{2})x^{2}+(a_{1}\oplus b_{1})x+(a_{0}\oplus b_{0})\end{aligned}}}

## Demonstration

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

### Step 1: polynomial multiplication

{\displaystyle {\begin{aligned}a(x)\bullet b(x)&=c(x)=(a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0})\bullet (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:

${\displaystyle c_{0}=a_{0}\bullet b_{0}}$
${\displaystyle c_{1}=a_{1}\bullet b_{0}\oplus a_{0}\bullet b_{1}}$
${\displaystyle c_{2}=a_{2}\bullet b_{0}\oplus a_{1}\bullet b_{1}\oplus a_{0}\bullet b_{2}}$
${\displaystyle c_{3}=a_{3}\bullet b_{0}\oplus a_{2}\bullet b_{1}\oplus a_{1}\bullet b_{2}\oplus a_{0}\bullet b_{3}}$
${\displaystyle c_{4}=a_{3}\bullet b_{1}\oplus a_{2}\bullet b_{2}\oplus a_{1}\bullet b_{3}}$
${\displaystyle c_{5}=a_{3}\bullet b_{2}\oplus a_{2}\bullet b_{3}}$
${\displaystyle c_{6}=a_{3}\bullet b_{3}}$

### Step 2: modular reduction

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

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

${\displaystyle x^{6}{\bmod {(}}x^{4}+1)=-x^{2}=x^{2}{\text{ over }}\operatorname {GF} (2^{8})}$
${\displaystyle x^{5}{\bmod {(}}x^{4}+1)=-x=x{\text{ over }}\operatorname {GF} (2^{8})}$
${\displaystyle x^{4}{\bmod {(}}x^{4}+1)=-1=1{\text{ over }}\operatorname {GF} (2^{8})}$

and from this we can deduce that ${\displaystyle x^{i}{\bmod {(}}x^{4}+1)=x^{i{\bmod {4}}}.}$

So

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

${\displaystyle d_{0}=c_{0}\oplus c_{4}}$
${\displaystyle d_{1}=c_{1}\oplus c_{5}}$
${\displaystyle d_{2}=c_{2}\oplus c_{6}}$
${\displaystyle d_{3}=c_{3}}$

### Step 3: matrix representation

If we use what was done in Step 1, the coefficient ${\displaystyle d_{3}}$, ${\displaystyle d_{2}}$, ${\displaystyle d_{1}}$ and ${\displaystyle d_{0}}$ can also be expressed as follows:

${\displaystyle d_{0}=a_{0}\bullet b_{0}\oplus a_{3}\bullet b_{1}\oplus a_{2}\bullet b_{2}\oplus a_{1}\bullet b_{3}}$
${\displaystyle d_{1}=a_{1}\bullet b_{0}\oplus a_{0}\bullet b_{1}\oplus a_{3}\bullet b_{2}\oplus a_{2}\bullet b_{3}}$
${\displaystyle d_{2}=a_{2}\bullet b_{0}\oplus a_{1}\bullet b_{1}\oplus a_{0}\bullet b_{2}\oplus a_{3}\bullet b_{3}}$
${\displaystyle d_{3}=a_{3}\bullet b_{0}\oplus a_{2}\bullet b_{1}\oplus a_{1}\bullet b_{2}\oplus a_{0}\bullet b_{3}}$

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

${\displaystyle d_{0}=2\bullet b_{0}\oplus 3\bullet b_{1}\oplus 1\bullet b_{2}\oplus 1\bullet b_{3}}$
${\displaystyle d_{1}=1\bullet b_{0}\oplus 2\bullet b_{1}\oplus 3\bullet b_{2}\oplus 1\bullet b_{3}}$
${\displaystyle d_{2}=1\bullet b_{0}\oplus 1\bullet b_{1}\oplus 2\bullet b_{2}\oplus 3\bullet b_{3}}$
${\displaystyle d_{3}=3\bullet b_{0}\oplus 1\bullet b_{1}\oplus 1\bullet b_{2}\oplus 2\bullet b_{3}}$

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:

${\displaystyle {\begin{bmatrix}d_{0}\\d_{1}\\d_{2}\\d_{3}\end{bmatrix}}={\begin{bmatrix}2&3&1&1\\1&2&3&1\\1&1&2&3\\3&1&1&2\end{bmatrix}}{\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{bmatrix}}}$

This can also be seen as the following:

${\displaystyle d_{0}=2b_{0}+3b_{1}+1b_{2}+1b_{3}}$
${\displaystyle d_{1}=1b_{0}+2b_{1}+3b_{2}+1b_{3}}$
${\displaystyle d_{2}=1b_{0}+1b_{1}+2b_{2}+3b_{3}}$
${\displaystyle d_{3}=3b_{0}+1b_{1}+1b_{2}+2b_{3}}$

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):

${\displaystyle {\begin{bmatrix}d_{0}\\d_{1}\\d_{2}\\d_{3}\end{bmatrix}}={\begin{bmatrix}14&11&13&9\\9&14&11&13\\13&9&14&11\\11&13&9&14\end{bmatrix}}{\begin{bmatrix}b_{0}\\b_{1}\\b_{2}\\b_{3}\end{bmatrix}}}$

Or:

${\displaystyle d_{0}=14b_{0}+11b_{1}+13b_{2}+9b_{3}}$
${\displaystyle d_{1}=9b_{0}+14b_{1}+11b_{2}+13b_{3}}$
${\displaystyle d_{2}=13b_{0}+9b_{1}+14b_{2}+11b_{3}}$
${\displaystyle d_{3}=11b_{0}+13b_{1}+9b_{2}+14b_{3}}$

## Test vectors for MixColumn(); not for InvMixColumn

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

## 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,
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,
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,
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,
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,
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,
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