How is multiplication performed in Galois Fields in the context of the AES algorithm?
In the context of the AES algorithm, multiplication in Galois Fields (GF) plays a important role in the encryption and decryption processes. The AES block cipher cryptosystem employs Galois Fields extensively to achieve its security objectives. To understand how multiplication is performed in Galois Fields within the AES algorithm, it is necessary to consider the concepts of Galois Fields, finite fields, and polynomial arithmetic.
Galois Fields, also known as finite fields, are mathematical structures that exhibit properties similar to those of ordinary arithmetic, but with a finite set of elements. The AES algorithm utilizes a specific Galois Field known as GF(2^8), which consists of 256 elements. Each element in this field can be represented as an 8-bit binary number, ranging from 00000000 to 11111111.
In GF(2^8), addition and subtraction are performed using the exclusive OR (XOR) operation, which is equivalent to binary addition without carry. Multiplication in GF(2^8) is more intricate and involves the use of irreducible polynomials. An irreducible polynomial is a polynomial of degree n that cannot be factored into the product of two lower-degree polynomials over GF(2^8).
To multiply two elements in GF(2^8), we utilize a multiplication algorithm known as the "carry-less multiplication" or "bitwise multiplication" algorithm. This algorithm is based on the concept of polynomial multiplication, where the binary representation of each element is treated as a polynomial.
Let's take two elements in GF(2^8), A and B, represented as binary numbers A = a7a6a5a4a3a2a1a0 and B = b7b6b5b4b3b2b1b0. To multiply A and B, we perform the following steps:
1. Initialize a result variable, R, to zero.
2. For each bit in B, starting from the least significant bit (b0):
a. If the current bit is 1, XOR R with A.
b. If the most significant bit of A is 1, left-shift A by one bit and XOR it with the irreducible polynomial, m(x).
c. Right-shift B by one bit.
The irreducible polynomial m(x) used in AES is x^8 + x^4 + x^3 + x + 1, which can be represented as 0x1B in hexadecimal notation.
Let's illustrate the multiplication of two elements, A = 10111001 and B = 00011110, in GF(2^8):
1. Initialize R = 00000000.
2. b0 = 0: No action required.
3. b1 = 1: XOR R with A, resulting in R = 10111001.
a. R = 00000000 XOR 10111001 = 10111001.
4. b2 = 1: XOR R with A, resulting in R = 00000001.
a. R = 10111001 XOR 10111001 = 00000000.
b. Left-shift A by one bit and XOR with m(x):
i. A = 01110010 XOR 00011011 = 01101001.
5. b3 = 1: XOR R with A, resulting in R = 01101001.
a. R = 00000000 XOR 01101001 = 01101001.
6. b4 = 1: XOR R with A, resulting in R = 01010010.
a. R = 01101001 XOR 01101001 = 01010010.
b. Left-shift A by one bit and XOR with m(x):
i. A = 11010010 XOR 00011011 = 11001001.
7. b5 = 0: No action required.
8. b6 = 0: No action required.
9. b7 = 0: No action required.
After performing all the steps, the final result R is 01010010, which corresponds to the product of A and B in GF(2^8).
It is important to note that multiplication in Galois Fields is not commutative, meaning that A * B may not be equal to B * A. Therefore, the order of the elements being multiplied is significant.
Multiplication in Galois Fields within the AES algorithm involves the use of irreducible polynomials and the carry-less multiplication algorithm. By treating the elements as polynomials and performing XOR and left-shift operations, the AES algorithm achieves multiplication in GF(2^8) to ensure the security of the encryption and decryption processes.
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