Homework 6
Due: Tuesday, November 4, 1997
Problem 1. Let
ksi be the primitive element of
GF(26)
defined by ksi = x mod x6
+ x + 1. Compute the orders of the elements
of ksii
for i=0,1, ... , 62.
Summarize your results in a log/order table. For which i's
are the ksii's
primitive? Do you see a pattern? Make a conjecture about this
pattern.
Problem 2. Consider
GF(33)
defined by the primitive polynomial p(x)
= x3 + 2x + 1, and let ksi
= x mod p(x). Find the minimum polynomial
m3(x)
of ksi3.
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 |
-
Last Modified: October 26, 1997