Rijndael MixColumns
From HandWiki
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 [math]\displaystyle{ b(x) = b_3 x^3 + b_2 x^2 + b_1 x + b_0 }[/math] which are elements within the field [math]\displaystyle{ \operatorname{GF}(2^8) }[/math]. The coefficients of the polynomials are elements within the prime sub-field [math]\displaystyle{ \operatorname{GF}(2) }[/math].
Each column is multiplied with a fixed polynomial [math]\displaystyle{ a(x) = 3x^3 + x^2 + x + 2 }[/math] modulo [math]\displaystyle{ x^4 + 1 }[/math]; the inverse of this polynomial is [math]\displaystyle{ a^{-1}(x) = 11x^3 + 13x^2 + 9x + 14 }[/math].
Demonstration
The polynomial [math]\displaystyle{ a(x) = 3x^3 + x^2 + x + 2 }[/math] will be expressed as [math]\displaystyle{ a(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0 }[/math].
Polynomial multiplication
- [math]\displaystyle{ \begin{align} a(x) \bullet b(x) = c(x) &= \left(a_3 x^3 + a_2 x^2 + a_1 x + a_0\right) \bullet \left(b_3 x^3 + b_2 x^2 + b_1 x + b_0\right) \\ &= 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{align} }[/math]
where:
- [math]\displaystyle{ \begin{align} c_0 &= a_0 \bullet b_0 \\ c_1 &= a_1 \bullet b_0 \oplus a_0 \bullet b_1 \\ c_2 &= a_2 \bullet b_0 \oplus a_1 \bullet b_1 \oplus a_0 \bullet b_2 \\ 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 \\ c_4 &= a_3 \bullet b_1 \oplus a_2 \bullet b_2 \oplus a_1 \bullet b_3 \\ c_5 &= a_3 \bullet b_2 \oplus a_2 \bullet b_3 \\ c_6 &= a_3 \bullet b_3 \end{align} }[/math]
Modular reduction
The result [math]\displaystyle{ c(x) }[/math] is a seven-term polynomial, which must be reduced to a four-byte word, which is done by doing the multiplication modulo [math]\displaystyle{ x^4 + 1 }[/math].
If we do some basic polynomial modular operations we can see that:
- [math]\displaystyle{ \begin{align} x^6 \bmod \left(x^4 + 1\right) &= -x^2 = x^2 \text{ over } \operatorname{GF}\left(2^8\right) \\ x^5 \bmod \left(x^4 + 1\right) &= -x = x \text{ over } \operatorname{GF}\left(2^8\right) \\ x^4 \bmod \left(x^4 + 1\right) &= -1 = 1 \text{ over } \operatorname{GF}\left(2^8\right) \end{align} }[/math]
In general, we can say that [math]\displaystyle{ x^i\bmod \left(x^4 + 1\right) = x^{i\bmod 4}. }[/math]
So
- [math]\displaystyle{ \begin{align} a(x) \otimes b(x) &= c(x) \bmod \left(x^4 + 1\right) \\ &= \left(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\right) \bmod \left(x^4 + 1\right) \\ &= 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 + \left(c_2 \oplus c_6\right) x^2 + \left(c_1 \oplus c_5\right) x + c_0 \oplus c_4 \\ &= d_3 x^3 + d_2 x^2 + d_1 x + d_0 \end{align} }[/math]
where
- [math]\displaystyle{ d_0 = c_0 \oplus c_4 }[/math]
- [math]\displaystyle{ d_1 = c_1 \oplus c_5 }[/math]
- [math]\displaystyle{ d_2 = c_2 \oplus c_6 }[/math]
- [math]\displaystyle{ d_3 = c_3 }[/math]
Matrix representation
The coefficient [math]\displaystyle{ d_3 }[/math], [math]\displaystyle{ d_2 }[/math], [math]\displaystyle{ d_1 }[/math] and [math]\displaystyle{ d_0 }[/math] can also be expressed as follows:
- [math]\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 }[/math]
- [math]\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 }[/math]
- [math]\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 }[/math]
- [math]\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 }[/math]
And when we replace the coefficients of [math]\displaystyle{ a(x) }[/math] with the constants [math]\displaystyle{ \begin{bmatrix}3&1&1&2\end{bmatrix} }[/math] used in the cipher we obtain the following:
- [math]\displaystyle{ d_0 = 2 \bullet b_0 \oplus 3 \bullet b_1 \oplus 1 \bullet b_2 \oplus 1 \bullet b_3 }[/math]
- [math]\displaystyle{ d_1 = 1 \bullet b_0 \oplus 2 \bullet b_1 \oplus 3 \bullet b_2 \oplus 1 \bullet b_3 }[/math]
- [math]\displaystyle{ d_2 = 1 \bullet b_0 \oplus 1 \bullet b_1 \oplus 2 \bullet b_2 \oplus 3 \bullet b_3 }[/math]
- [math]\displaystyle{ d_3 = 3 \bullet b_0 \oplus 1 \bullet b_1 \oplus 1 \bullet b_2 \oplus 2 \bullet b_3 }[/math]
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:
- [math]\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} }[/math]
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):
- [math]\displaystyle{ \begin{bmatrix}b_0\\b_1\\b_2\\b_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}d_0\\d_1\\d_2\\d_3\end{bmatrix} }[/math]
Or:
- [math]\displaystyle{ \begin{align} b_0 &= 14 \bullet d_0 \oplus 11 \bullet d_1 \oplus 13 \bullet d_2 \oplus 9 \bullet d_3 \\ b_1 &= 9 \bullet d_0 \oplus 14 \bullet d_1 \oplus 11 \bullet d_2 \oplus 13 \bullet d_3 \\ b_2 &= 13 \bullet d_0 \oplus 9 \bullet d_1 \oplus 14 \bullet d_2 \oplus 11 \bullet d_3 \\ b_3 &= 11 \bullet d_0 \oplus 13 \bullet d_1 \oplus 9 \bullet d_2 \oplus 14 \bullet d_3 \end{align} }[/math]
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)
See also
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