<s>[INST] Polynomial Multiplication in GF(256) I would like to compute the following:
10100011 * 01100011 in GF(256) using the AES irreducible polynomial. 

So first we get the polynomials:
10100011 = x^8 + x^6 + x + 1
01100011 = x^7 + x^6 + x^2 + 1

Multiplying these out, I got the result:
x^56 + x^48 + x^42 + x^36 + x^16 + x^12 + x^3 + x^2 + x + 1

So when I try to reduce this using the AES irreducible polynomial, x^8 + x^4 + x^3 + x + 1, I am getting the result:
 x^7 + x^6 with a remainder of x^42 + x^36 + x^28 + x^24  +x^21 + x^18 + x^16 + x^12 + x^8 + x^6 + x^3 + x^2 + x + 1

This cannot be correct, but I'm having trouble finding where I made a mistake. I have a feeling my multiplication of the two polynomials is incorrect. 
 [/INST] The exponentiation law is: 
x^a * x^b = x^(a+b)
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