Question 3 [Points 21]

A. (Points 6) In the context of prime field GF(13), (i) What are the elements of this prime field? (ii) What is the additive inverse of 9 in this field? (iii) What is the multiplicative inverse of 9 in this field?

B. (Points 9) In the context of extension field GF(2⁵), (i) Represent A(x) = 15 in the above field. (Note: It will be a polynomial) (ii) Represent B(x) = 22 in the same field. (iii) Determine A(x)+B(x) (as a polynomial) and express as an integer.

C. (Points 6) Using Table 4.2, of GF(2⁸), determine the multiplicative inverses of 3D (hexa) (Use Table 4.2 in the textbook) . Also, determine the AES S-box substitutions for this 8-bit string (use Table 4.3 in the textbook).