modulo reduction in field of non primitive order

Question

I'm getting started with AES and need to calculate the inverse for S-byte table. I'm trying to generate exponential and logarithm table for inversion with generator 3. The exponents works fine till I reach 3 ^ 8 which is 0xff. Exponential table

 0x03 0x05 0x0f 0x11 0x33 0x55 0xff 0x1a 0x2e 0x72 0x96 0xa1 0xf8 0x13 0x35 0x5f 
0xe1 0x38 0x48 0xd8 0x73 0x95 0xa4 0xf7 0x02 0x06 0x0a 0x1e 0x22 0x66 0xaa 0xe5 
0x34 0x5c 0xe4 0x37 0x59 0xeb 0x26 0x6a 0xbe 0xd9 0x70 0x90 0xab 0xe6 0x31 0x53 
0xf5 0x04 0x0c 0x14 0x3c 0x44 0xcc 0x4f 0xd1 0x68 0xb8 0xd3 0x6e 0xb2 0xcd 0x4c 
0xd4 0x67 0xa9 0xe0 0x3b 0x4d 0xd7 0x62 0xa6 0xf1 0x08 0x18 0x28 0x78 0x88 0x83 
0x9e 0xb9 0xd0 0x6b 0xbd 0xdc 0x7f 0x81 0x98 0xb3 0xce 0x49 0xdb 0x76 0x9a 0xb5 
0xc4 0x57 0xf9 0x10 0x30 0x50 0xf0 0x0b 0x1d 0x27 0x69 0xbb 0xd6 0x61 0xa3 0xfe 
0x19 0x2b 0x7d 0x87 0x92 0xad 0xec 0x2f 0x71 0x93 0xae 0xe9 0x20 0x60 0xa0 0xfb 
0x16 0x3a 0x4e 0xd2 0x6d 0xb7 0xc2 0x5d 0xe7 0x32 0x56 0xfa 0x15 0x3f 0x41 0xc3 
0x5e 0xe2 0x3d 0x47 0xc9 0x40 0xc0 0x5b 0xed 0x2c 0x74 0x9c 0xbf 0xda 0x75 0x9f 
0xba 0xd5 0x64 0xac 0xef 0x2a 0x7e 0x82 0x9d 0xbc 0xdf 0x7a 0x8e 0x89 0x80 0x9b 
0xb6 0xc1 0x58 0xe8 0x23 0x65 0xaf 0xea 0x25 0x6f 0xb1 0xc8 0x43 0xc5 0x54 0xfc 
0x1f 0x21 0x63 0xa5 0xf4 0x07 0x09 0x1b 0x2d 0x77 0x99 0xb0 0xcb 0x46 0xca 0x45 
0xcf 0x4a 0xde 0x79 0x8b 0x86 0x91 0xa8 0xe3 0x3e 0x42 0xc6 0x51 0xf3 0x0e 0x12 
0x36 0x5a 0xee 0x29 0x7b 0x8d 0x8c 0x8f 0x8a 0x85 0x94 0xa7 0xf2 0x0d 0x17 0x39 
0x4b 0xdd 0x7c 0x84 0x97 0xa2 0xfd 0x1c 0x24 0x6c 0xb4 0xc7 0x52 0xf6 0x01 0x03 

I'm having trouble to calculate past 255 involving modulo reduction. I'm assuming the Mod here is 256 since the largest value is 255. The value I generate after 0xff(255) * 3 = 257. How do I perform a calculation to get 0x1a?

Thank you.

Answer 1

The polynomial used for AES = x^8 + x^4 + x^3 + x + 1 (0x11B) is prime, but not primitive. All non-zero numbers can be considered to be a power of 3.

3 raised to the 8th power modulo 0x11B = 3 * 0xFF modulo 0x11B = 0x101 modulo 0x11B = 0x1A, which is what you have in your table. 0x03^0xFE%0x11B = 0xF6, also in your table.