Addition in GF(2^4): Compute addition A(x)+B(x) mod P(x) in GF(2^4) using the irreducible polynomial P(x) =...

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Addition in GF(2^4): Compute addition A(x)+B(x) mod P(x) in GF(2^4) using the irreducible polynomial P(x) = x^4+x+1. What is the influence of the choice of the reduction polynomial on the computation of addition? -A(x) = x 2+1, B(x)=x^3+x^2+1 - A(x) = x^2+1, B(x)=x+1 2. Multiplication in GF(2^4): Compute multiplication A(x)*B(x) mod P(x) in GF (2^4) using the irreducible polynomial P(x)=x^4 +x-1 (show all the steps for both examples). What is the influence of the choice of the reduction polynomial on the computation of multiplication? -A(x)=x^2+1, B(x)=x^3+x^2+1 -A(x)=x^2+1, B(x)=x+1 The MixColumns transformation of AES consists of a matrix-vector multiplication in the field GF(2^8) with irreducible polynomial P(x) = x^8+x^4+x^3+x+1. Let b = (b7x7+...+b0) be one of the (four) input bytes to the vector-matrix multiplication. Each input byte is multiplied with the constants 01, 02 and 03. Your task is to provide exact equations for computing those three constant multiplications after reductions. We denote the result by d = (d7x7+...-d0). 1. Equations for computing the 8 bits of d = 01 - b mod P(x). 2. Equations for computing the 8 bits of d = 02-b mod P(x). 3. Equations for computing the 8 bits of d = 03-b mod P(x). Note: The AES specification uses "01" to represent the polynomial 1, "02" to represent the polynomial x, and "03" to represent x+1. We recall from the discussion of stream ciphers that a 2-input XOR gate performs a GF(2) addition. How many 2-input XOR gates are required to perform one constant multiplication by 01, 02 and 03, respectively, in GF(2^8)? The InvMixColumns transformation of AES consists of a matrix-vector multiplication in the field GF(2^8) with irreducible polynomial P(x) = x^8+x^4+x^3+x+1 as well during the decryption of AES. Let b = (b7x7+...+b0) be one of the (four) input bytes to the vector- matrix multiplication. Each input byte is multiplied with the constants OE, OB, OD and 09. Your task is to provide exact equations for computing those four constant multiplications after reductions. We denote the result by d = (d7x7+...+ d0). Addition in GF(2^4): Compute addition A(x)+B(x) mod P(x) in GF(2^4) using the irreducible polynomial P(x) = x^4+x+1. What is the influence of the choice of the reduction polynomial on the computation of addition? -A(x) = x 2+1, B(x)=x^3+x^2+1 - A(x) = x^2+1, B(x)=x+1 2. Multiplication in GF(2^4): Compute multiplication A(x)*B(x) mod P(x) in GF (2^4) using the irreducible polynomial P(x)=x^4 +x-1 (show all the steps for both examples). What is the influence of the choice of the reduction polynomial on the computation of multiplication? -A(x)=x^2+1, B(x)=x^3+x^2+1 -A(x)=x^2+1, B(x)=x+1 The MixColumns transformation of AES consists of a matrix-vector multiplication in the field GF(2^8) with irreducible polynomial P(x) = x^8+x^4+x^3+x+1. Let b = (b7x7+...+b0) be one of the (four) input bytes to the vector-matrix multiplication. Each input byte is multiplied with the constants 01, 02 and 03. Your task is to provide exact equations for computing those three constant multiplications after reductions. We denote the result by d = (d7x7+...-d0). 1. Equations for computing the 8 bits of d = 01 - b mod P(x). 2. Equations for computing the 8 bits of d = 02-b mod P(x). 3. Equations for computing the 8 bits of d = 03-b mod P(x). Note: The AES specification uses "01" to represent the polynomial 1, "02" to represent the polynomial x, and "03" to represent x+1. We recall from the discussion of stream ciphers that a 2-input XOR gate performs a GF(2) addition. How many 2-input XOR gates are required to perform one constant multiplication by 01, 02 and 03, respectively, in GF(2^8)? The InvMixColumns transformation of AES consists of a matrix-vector multiplication in the field GF(2^8) with irreducible polynomial P(x) = x^8+x^4+x^3+x+1 as well during the decryption of AES. Let b = (b7x7+...+b0) be one of the (four) input bytes to the vector- matrix multiplication. Each input byte is multiplied with the constants OE, OB, OD and 09. Your task is to provide exact equations for computing those four constant multiplications after reductions. We denote the result by d = (d7x7+...+ d0).

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1 Addition in GF24 Given irreducible polynomial Px x4 x 1 and two polynomials Ax x2 1 Bx x3 x2 1 we need to compute the addition Ax Bx mod Px in GF24 Step 1 Perform the subtraction Ax Bx Ax Bx x2 1 x3 ... View the full answer