the MixColumns operation in AES uses the polynomial c(y) = (03)y^3 + (01)y^2 + (01)y + (02) ,

where the coefficients of c(y) are bytes written in hexadecimal (i.e. base 16) notation. Arithmetic involving this polynomial requires the computation of products involving the bytes (01), (02) and (03) in the AES field GF(28 ).

(a) Write the values (01), (02), (03) as their respective polynomial representatives c1(x), c2(x) and c3(x).

(b) Let b = (b7 b6 b1 b0) be any byte.

(i) Let d = (01)b be the product of the bytes (01) and b in the AES field GF(28 ) (with arithmetic modulo m(x) = x^8 + x^4 + x^3 + x + 1), and write d = (d7 d6 d1 d0). Provide exact equations for the bits di , 0 i 7, in terms of the bits bi of b.

(ii)Provide analogous equations as in part (i) for the byte product d = (02)b.

(iii) Provide analogous equations as in part (i) for the byte product d = (03)b.