Question

This question concerns the field GF(256). The modulus is P(x)=x⁸ +x⁴+x³+x+1. Please answer the following questions about arithmetic in this field. a) If p(x)=x⁴+x³+x, what is the corresponding hexadecimal representation? Please prepend an "0x" on your answer and use lowercase letters (if letters occur)_____________.

b) If p(x)=x⁷+x ⁶+x⁵+x⁴+x, what is the corresponding hexadecimal representation? Please prepend an "0x" on your answer and use lowercase letters (if letters occur)___________.
c) Consider the hex byte 0x25. What is the corresponding element of GF(256) (as a polynomial)? ____________
d) Consider the hex byte 0x2e. What is the corresponding element of GF(256) (as a polynomial)?_________

Answer 1

In GF(256), the hexadecimal representation of p(x) = x⁴ + x³ + x is 0xD, and the hexadecimal representation of p(x) = x⁷ + x⁶ + x⁵ + x⁴ + x is 0x1D. The corresponding element of GF(256) for the hex byte 0x25 is x⁵ + x² + 1, and for the hex byte 0x2E is x⁵ + x⁴ + x³ + x + 1.

Step-by-step explanation:

a) To convert p(x) = x⁴ + x³ + x to hexadecimal representation, we can group the coefficients in sets of 4 from the right. Starting from the right, we have x³, x², x¹, and x⁰, which correspond to 1101. Converting 1101 to hexadecimal gives us D. Thus, the hexadecimal representation of p(x) is 0xD.

b) Similarly, to convert p(x) = x⁷ + x⁶ + x⁵ + x⁴ + x to hexadecimal representation, we group the coefficients in sets of 4 from the right. Starting from the right, we have x⁶, x⁵, x⁴, x³, and x², which correspond to 11101. Converting 11101 to hexadecimal gives us 1D. Thus, the hexadecimal representation of p(x) is 0x1D.

c) To find the corresponding element of GF(256) for the hex byte 0x25, we convert it to binary, which gives us 0010 0101. We can then interpret this binary representation as the coefficients of a polynomial in GF(256), which gives us x⁵ + x² + 1.

d) Similarly, to find the corresponding element of GF(256) for the hex byte 0x2E, we convert it to binary, which gives us 0010 1110. Interpreting this binary representation as the coefficients of a polynomial in GF(256), we get x⁵ + x⁴ + x³ + x + 1.

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