Question

Problem 4. Consider the field Z2[x]/(F), where $ = x5 + x2 + 1. In this field, we write abcde as a notation for ax4 + bx3 + cx2 + dx +e, where a, b, c, d, e are elements of Z2. For example, 11010 is a notation for the element 1x4 + 1x3 + 0x2 + 1x+0 = x4 + x3 + x. Compute the following. Make sure to write all of your answers either as polynomials of degree less than 5, or as a sequence of 5 binary digits. (a) 11010 + 01111. (b) 11010 · 01111. (c) 11010-1.

Answer 1

wed 12 13 14 1. Here, abcde = a.xte bx² pcx²+date a, b, c, d, e, t7z. a). From above notation, 11010 = 1.x4 + 1.223 +0.x"+1X 0.1 = x4 &n3+x. Also, OLLLL = 0.x4 + 1x² + b.x² tlonta = x3 + x²+x+1. Now, 12020 + OLILL = (nt tn3+x) + (x²+x²+x+1) = (x4 + 2x² + x²+2x+2) moda = x4 +0.23 +100% +0.x+1 11010 01111 = 10101 19 1 dorp

Here 11010 = x++* +. 01111 = x++z2+ 2+2. Therefore, £2010. oll1 = (44+ n²+ n. (a3 + x² entir- = x² + x + x statt notastat - + x² + x4 + x3 + x2+x. = (+ + 2x6+ 2.35+ 2x4+ 2.3} + x²+x.) moda = x² + x²+x ) = x² [ x S + x² +1] + +64+a) = x² 0 + (a4tx) atta 11010. 01/11 - 01001 ) Note): n&ta²+1 is a zero element or additive identity of the given field. I

#©: (11010) = (n48934 m) Toate:- Page: By Euclidean Algorithm - (over 72) x5 + x2 + 1 = r. (nt + x3 + x)+(24+1) = x4 +3+ x = 1. (24 + 2) + (x3 fatt) 7 x4+1 = x(x3 + x++) + (x2+x+2) = x²+x+1 = x (x²+x+6) + (22+4) z (224 at 1) = 1. (x2++) + x = x2 + t = nort1. Crtol

By backward substitution- 1 = ( 2+1) - 7. - (1441) - ndcnatu+1) - (2+134 = (1+2). (2²+1) -a(0²1211) = (1+r) (mt*+1) - (nan+1) 4 (²741) = (1+x) (x3 +*+1) – (2+ **+ x) (x2+x+1) = (1+2) (x34x+1) - x2(x²+2+1). = (1+0) (M3+*+1) -- **{(2441) -- ~(~34901) = (1+x+23) (1424x3) - x² (2411) = (1+ 2+ 3) & (24px 3px) -(24+1)} xenters = (Itatn3) (a4fx3 + 2) + (1+2+x²+x3). ca 1 = (1+a+23) ( 241 23 fn) - Citat a tas) 565 +x+1) – 4- (n*e x strony = (1+x+ 23 + x + x2 + x3 p 24 ) ( 24 p 231 2) - that 2² +23) ( 25 1841)

Date: Page: Bell to 1 = (1+2x+223 p nt+22) (atene ( As inginer friels 1 = (1+x²124) (nap n3px). I (ate n²+1 = (1ea²ent) = (11010) = ( 101011