# Homework 5

Problem 1.  Consider  GF(33)  defined by the primitive polynomial  p(x) = x3 + 2x + 1, and let  ksi = x mod p(x).  Find the minimum polynomial  m5(x)  of  ksi5.   You may assume the following theorems:

• ap = a  for all  a  in  GF(p) for  p  a prime integer.
• (Sumj=1n aj)p = Sumj=1n (aj)p  in any field of characteristic  p.

You may use the following table for you calculations:

 GF(33) defined by the primitive polynomial  p(x) = x3 + 2x + 1 Antilog Log Antilog Log 000 -INF 100 0 200 13 010 1 020 14 001 2 002 15 210 3 120 16 021 4 012 17 212 5 121 18 111 6 222 19 221 7 112 20 202 8 101 21 110 9 220 22 011 10 022 23 211 11 122 24 201 12 102 25

Problem 2.
You will find below the Antilog/Log table of  GF(26)  based on the primitive polynomial

p(x) = x6 + x + 1

Use this table to compute the minimum polynomial  m5(x)  of  ksi5 , where  ksi  is the primitive element defined by  p(x).

 Antilog Log Antilog Log Antilog Log Antilog Log 000000 -INF 000101 15 101001 31 111001 47 100000 0 110010 16 100100 32 101100 48 010000 1 011001 17 010010 33 010110 49 001000 2 111100 18 001001 34 001011 50 000100 3 011110 19 110100 35 110101 51 000010 4 001111 20 011010 36 101010 52 000001 5 110111 21 001101 37 010101 53 110000 6 101011 22 110110 38 111010 54 011000 7 100101 23 011011 39 011101 55 001100 8 100010 24 111101 40 111110 56 000110 9 010001 25 101110 41 011111 57 000011 10 111000 26 010111 42 111111 58 110001 11 011100 27 111011 43 101111 59 101000 12 001110 28 101101 44 100111 60 010100 13 000111 29 100110 45 100011 61 001010 14 110011 30 010011 46 100001 62

Problem 3.

• (a) Draw the linear sequential circuit (LSC) that multiplies by the polynomial

h(x) = 1 + x3 +x6

• (b) Draw the linear sequential circuit (LSC) that divides by the polynomial

g(x) = 1 + x2 + x4 + x6 + x7

• (c) Draw the linear sequential circuit (LSC) that simultaneously multiplies by h(x)  and divides by  g(x).

Problem 4.  Draw an LSC which takes as inputs polynomials  a(x)  and  b(x)  and then produces the output  h(x)a(x) + k(x)b(x), where  h(x)  and  k(x)  are the polynomials:

h(x) = 1 + x4 + x10       and       k(x) = x + x2 + x4 + x7 + x9