1. The Advanced Encryption Standard Part 2: Mathematical Background
CSCI 5857: Encoding and Encryption

2. Outline Modular multiplicative inverses Galois Field mathematics
Galois Field inverses
Uses in AES

3. Mathematical Goals
S-Boxes and other transformations should have mathematical basis
Can insure useful properties (nonlinearity, etc.)
Can re-derive as needed for larger keys
Mapping should appear “random” (no simple patterns between inputs and outputs)

4. Modular Multiplication
a  b mod m = remainder left after (a  b)/m
Example: multiplication table mod 7

5. Modular Multiplicative Inverses
b is inverse of a mod m if ab mod m = 1 (b = a -1 mod m)
Example: 5 = 3-1 mod 7 since 3 x 5 = 15 = 1 mod 7
Creates nonlinear “pseudorandom” mappings a
a -1
none 1 2 4 3 5 6

6. Lack of Multiplicative Inverses
Problem: Only works if m is a prime number Otherwise, some numbers have no inverse
Example: modular inverses mod 8 a
a -1
none 1 2 3 4 5 6 7

7. Galois Fields for Inverses
Goal: use this idea in cases where m = 2n (that is, m is the size of a typical block)
Galois Fields
Represent byte to transform as a polynomial
Compute inverse of that polynomial mod some other “prime” polynomial
Galois Field with m = 28 used to create S-Boxes for AES , mapping 256 possible byte inputs to 256 possible byte outputs

8. Galois Field Mathematics
Step 1: Represent binary numbers with n bits as polynomial of degree n
Example: n = 3 GF(23)
Binary
Polynomial
000
0x2 + 0x + 0
001
0x2 + 0x + 1 1
010
0x2 + 1x + 0 x
011
0x2 + 1x + 1
x + 1
100
1x2 + 0x + 0 x2
101
1x2 + 0x + 1
x2 + 1
110
1x2 + 1x + 0
x2 + x
111
1x2 + 1x + 1
x2 + x + 1

9. Galois Field Mathematics (1)
All coefficients are binary (1 or 0)
Addition/subtraction in mod 2 = XOR function
Examples:
x2 + x + 1
x + 1
x2 + 2x + 2
= x2 + 0x + 0 = x2
since 2 mod 2 = 0
x2 - (x + 1) x2 - x – 1 = x2 + x + 1 since -1 mod 2 = 1

10. Galois Field Mathematics (2)
Step 2: Find a “prime” polynomial Pn of degree n
Not a multiple of any two other polynomials (other than 1 and itself)
Example for GF(23): P3 = x3 + x + 1
Used in AES for GF(28): P8 = x8 + x4 + x3 + x + 1

11. Galois Field Mathematics (3)
Step 3: Compute multiplication table for all pairs of polynomials Pi x Pj mod Pn
Will need to compute mod if order of Pi x Pj is k  n
Simple (inefficient) way: compute Pi x Pj – xk-nPn
Example for GF(23):

12. Galois Field Example Example: Multiplying 110 and 101
110  x2 + x 011  x + 1
(x2 + x)(x + 1) = x3 + 2x2 + x = x3 + x 2 mod 2 = 0
(x3 + x) mod (x3 + x + 1) = x3 + x x3 + x = mod 2 = 1

13. Galois Field Inverses
Inverse b-1 of a binary number b in GF(2n) b-1 x b = 1 in GF(2n)
Example: GF(23) b
000
001
010
011
100
101
110
111
b-1
none

14. Galois Fields in AES SubBytes stage MixColumns Stage
AES mathematics based on GF(28)
Prime polynomial = x8 + x4 + x3 + x + 1
SubBytes stage
Basis of S-Boxes
MixColumns Stage
Uses matrix multiplication in GF(28)
Round Key Generation
Adds extra “random” bits to each round key

15. What’s Next Let me know if you have any questions
Continue on to the next lecture on AES: Mathematical Backgorund