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Problem 4. Consider the field Z2[x]/(F), where $ = x5 + x2 + 1. In this...

Problem 4. Consider the field Z2[x]/(F), where $ = x5 + x2 + 1. In this field, we write abcde as a notation for ax4 + bx3 + c

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.
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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+1XHere 11010 = x++* +. 01111 = x++z2+ 2+2. Therefore, £2010. oll1 = (44+ n²+ n. (a3 + x² entir- = x² + x + x statt notastat - +#©: (11010) = (n48934 m) Toate:- Page: By Euclidean Algorithm - (over 72) x5 + x2 + 1 = r. (nt + x3 + x)+(24+1) = x4 +3+ x =By backward substitution- 1 = ( 2+1) - 7. - (1441) - ndcnatu+1) - (2+134 = (1+2). (2²+1) -a(0²1211) = (1+r) (mt*+1) - (nan+1)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) = (11

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Problem 4. Consider the field Z2[x]/(F), where $ = x5 + x2 + 1. In this...
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