1 - A Program for Computing Primitive Polynomials.
2
3Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
4GNU General Public License. This is free software, and you are welcome
5to redistribute it under certain conditions; see the GNU General Public License
6for details.
7
8Self-check passes...
9
10
11Primitive polynomial modulo 2 of degree 2
12
13x ^ 2 + x + 1, 2
14
15
16 - A Program for Computing Primitive Polynomials.
17
18Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
19GNU General Public License. This is free software, and you are welcome
20to redistribute it under certain conditions; see the GNU General Public License
21for details.
22
23Self-check passes...
24
25
26Primitive polynomial modulo 2 of degree 17
27
28x ^ 17 + x ^ 3 + 1, 2
29
30
31 - A Program for Computing Primitive Polynomials.
32
33Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
34GNU General Public License. This is free software, and you are welcome
35to redistribute it under certain conditions; see the GNU General Public License
36for details.
37
38Self-check passes...
39
40
41Primitive polynomial modulo 2 of degree 43
42
43x ^ 43 + x ^ 6 + x ^ 4 + x ^ 3 + 1, 2
44
45
46 - A Program for Computing Primitive Polynomials.
47
48Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
49GNU General Public License. This is free software, and you are welcome
50to redistribute it under certain conditions; see the GNU General Public License
51for details.
52
53Self-check passes...
54
55
56Primitive polynomial modulo 2 of degree 62
57
58x ^ 62 + x ^ 6 + x ^ 5 + x ^ 3 + 1, 2
59
60
61 - A Program for Computing Primitive Polynomials.
62
63Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
64GNU General Public License. This is free software, and you are welcome
65to redistribute it under certain conditions; see the GNU General Public License
66for details.
67
68Self-check passes...
69
70
71Primitive polynomial modulo 3 of degree 30
72
73x ^ 30 + x + 2, 3
74
75
76 - A Program for Computing Primitive Polynomials.
77
78Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
79GNU General Public License. This is free software, and you are welcome
80to redistribute it under certain conditions; see the GNU General Public License
81for details.
82
83Self-check passes...
84
85
86Primitive polynomial modulo 5 of degree 22
87
88x ^ 22 + x ^ 2 + x + 3, 5
89
90
91 - A Program for Computing Primitive Polynomials.
92
93Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
94GNU General Public License. This is free software, and you are welcome
95to redistribute it under certain conditions; see the GNU General Public License
96for details.
97
98Self-check passes...
99
100
101Primitive polynomial modulo 7 of degree 16
102
103x ^ 16 + 2 x + 3, 7
104
105
106 - A Program for Computing Primitive Polynomials.
107
108Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
109GNU General Public License. This is free software, and you are welcome
110to redistribute it under certain conditions; see the GNU General Public License
111for details.
112
113Self-check passes...
114
115
116Primitive polynomial modulo 17 of degree 14
117
118x ^ 14 + x ^ 2 + 2 x + 6, 17
119
120
121 - A Program for Computing Primitive Polynomials.
122
123Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
124GNU General Public License. This is free software, and you are welcome
125to redistribute it under certain conditions; see the GNU General Public License
126for details.
127
128Self-check passes...
129
130
131Primitive polynomial modulo 137 of degree 7
132
133x ^ 7 + x + 3, 137
134
135
136 - A Program for Computing Primitive Polynomials.
137
138Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
139GNU General Public License. This is free software, and you are welcome
140to redistribute it under certain conditions; see the GNU General Public License
141for details.
142
143Self-check passes...
144
145
146Primitive polynomial modulo 223 of degree 8
147
148x ^ 8 + x + 51, 223
149
150
151 - A Program for Computing Primitive Polynomials.
152
153Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
154GNU General Public License. This is free software, and you are welcome
155to redistribute it under certain conditions; see the GNU General Public License
156for details.
157
158Self-check passes...
159
160
161Primitive polynomial modulo 557 of degree 6
162
163x ^ 6 + x + 5, 557
164
165
166 - A Program for Computing Primitive Polynomials.
167
168Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
169GNU General Public License. This is free software, and you are welcome
170to redistribute it under certain conditions; see the GNU General Public License
171for details.
172
173Self-check passes...
174
175
176Primitive polynomial modulo 2 of degree 4
177
178x ^ 4 + x + 1, 2
179
180Confirming polynomial is primitive with a brute force check.
181WARNING: You may wait an impossibly long time!
182To abort, hit control-C in your console window to stop this program.
183x ^ 4 + x + 1, 2 confirmed primitive!
184+--------- OperationCount --------------------------------
185|
186| Integer factorization: Table lookup + Trial division + Pollard Rho
187|
188| Number of trial divisions : 0
189| Number of gcd's computed : 0
190| Number of primality tests : 0
191| Number of squarings: 0
192|
193| Polynomial Testing
194|
195| Total num. degree 4 poly mod 2 : 16
196| Number of possible primitive poly: 2
197| Polynomials tested : 4
198| Const. coeff. was primitive root : 2
199| Free of linear factors : 1
200| Irreducible to power >=1 : 1
201| Had order r (x^r = integer) : 1
202| Passed const. coeff. test : 1
203| Had order m (x^m != integer) : 1
204|
205+-----------------------------------------------------
206
207
208 - A Program for Computing Primitive Polynomials.
209
210Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
211GNU General Public License. This is free software, and you are welcome
212to redistribute it under certain conditions; see the GNU General Public License
213for details.
214
215Self-check passes...
216
217
218Primitive polynomial modulo 3 of degree 5
219
220x ^ 5 + 2 x + 1, 3
221
222Confirming polynomial is primitive with a brute force check.
223WARNING: You may wait an impossibly long time!
224To abort, hit control-C in your console window to stop this program.
225x ^ 5 + 2 x + 1, 3 confirmed primitive!
226+--------- OperationCount --------------------------------
227|
228| Integer factorization: Table lookup + Trial division + Pollard Rho
229|
230| Number of trial divisions : 0
231| Number of gcd's computed : 0
232| Number of primality tests : 0
233| Number of squarings: 0
234|
235| Polynomial Testing
236|
237| Total num. degree 5 poly mod 3 : 243
238| Number of possible primitive poly: 22
239| Polynomials tested : 8
240| Const. coeff. was primitive root : 3
241| Free of linear factors : 1
242| Irreducible to power >=1 : 1
243| Had order r (x^r = integer) : 1
244| Passed const. coeff. test : 1
245| Had order m (x^m != integer) : 1
246|
247+-----------------------------------------------------
248
249
250 - A Program for Computing Primitive Polynomials.
251
252Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
253GNU General Public License. This is free software, and you are welcome
254to redistribute it under certain conditions; see the GNU General Public License
255for details.
256
257Self-check passes...
258
259
260There are 6 primitive polynomials modulo 2 of degree 5
261
262
263
264Primitive polynomial modulo 2 of degree 5
265
266x ^ 5 + x ^ 2 + 1, 2
267
268
269
270Primitive polynomial modulo 2 of degree 5
271
272x ^ 5 + x ^ 3 + 1, 2
273
274
275
276Primitive polynomial modulo 2 of degree 5
277
278x ^ 5 + x ^ 3 + x ^ 2 + x + 1, 2
279
280
281
282Primitive polynomial modulo 2 of degree 5
283
284x ^ 5 + x ^ 4 + x ^ 2 + x + 1, 2
285
286
287
288Primitive polynomial modulo 2 of degree 5
289
290x ^ 5 + x ^ 4 + x ^ 3 + x + 1, 2
291
292
293
294Primitive polynomial modulo 2 of degree 5
295
296x ^ 5 + x ^ 4 + x ^ 3 + x ^ 2 + 1, 2
297
298
299 - A Program for Computing Primitive Polynomials.
300
301Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
302GNU General Public License. This is free software, and you are welcome
303to redistribute it under certain conditions; see the GNU General Public License
304for details.
305
306Error. Polynomial has bad syntax or coefficients are out of range. [ Error in parser converting polynomial from string x 1 for p = 2 Expecting to see x^ or x or x ^ integer in sentence x 1 ]
307This program generates primitive polynomials of degree n modulo p.
308
309Usage: Generate a single random polynomial of degree n modulo p where p is a prime >= 2 and n is an integer >= 2
310 Primpoly p n
311Example:
312 Primpoly 2 4
313 Self-check passes...
314 Primitive polynomial modulo 2 of degree 4
315 x ^ 4 + x + 1, 2
316Usage: Test whether a polynomial is primitive modulo p.
317 Primpoly -t <Polynomial to test>, p
318 If you leave off the modulus p we default to p = 2
319Examples:
320 Primpoly -t x^4 + x + 1, 2
321 Self-check passes...
322 x ^ 4 + x + 1, 2 is primitive!
323
324 Primpoly -t x^4 + x + 1
325 Self-check passes...
326 x ^ 4 + x + 1, 2 is primitive!
327Usage: Generate all primitive polynomial of degree n modulo p.
328 Primpoly -a p n
329Example:
330 Primpoly -a 2 4
331 Self-check passes...
332 Primitive polynomial modulo 2 of degree 4
333 x ^ 4 + x + 1, 2
334 Primitive polynomial modulo 2 of degree 4
335 x ^ 4 + x ^ 3 + 1, 2
336Usage: Same but show computation statistics.
337 Primpoly -s p n
338Examples:
339
340 Primpoly.exe -s 13 19
341 Self-check passes...
342 Primitive polynomial modulo 13 of degree 19
343 x ^ 19 + 9 x + 2, 13
344
345 +--------- OperationCount --------------------------------
346 |
347 | Integer factorization: Table lookup + Trial division + Pollard Rho
348 |
349 | Number of trial divisions : 0
350 | Number of gcd's computed : 9027
351 | Number of primality tests : 2
352 | Number of squarings: 9026
353 |
354 | Polynomial Testing
355 |
356 | Total num. degree 19 poly mod 13 : 1461920290375446110677
357 | Number of possible primitive poly: 6411930599771980992
358 | Polynomials tested : 120
359 | Const. coeff. was primitive root : 46
360 | Free of linear factors : 11
361 | Irreducible to power >=1 : 1
362 | Had order r (x^r = integer) : 1
363 | Passed const. coeff. test : 1
364 | Had order m (x^m != integer) : 1
365 |
366 +-----------------------------------------------------
367Usage: Print help message.
368 Primpoly -h
369 <Prints this help message.>
370
371
372Primitive polynomials find many uses in mathematics and communications
373engineering:
374 * Generation of pseudonoise (PN) sequences for spread spectrum
375 communications and chip fault testing.
376 * Generating Sobol sequences for high dimensional numerical integration.
377 * Generation of CRC and Hamming codes.
378 * Generation of Galois (finite) fields for use in decoding Reed-Solomon
379 and BCH error correcting codes.
380
381For detailed technical information, see my web page
382 http://seanerikoconnor.freeservers.com/Mathematics/AbstractAlgebra/PrimitivePolynomials/overview.html
383
384
385 - A Program for Computing Primitive Polynomials.
386
387Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
388GNU General Public License. This is free software, and you are welcome
389to redistribute it under certain conditions; see the GNU General Public License
390for details.
391
392Self-check passes...
393x ^ 4 + x + 1, 2 is primitive!
394
395 - A Program for Computing Primitive Polynomials.
396
397Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
398GNU General Public License. This is free software, and you are welcome
399to redistribute it under certain conditions; see the GNU General Public License
400for details.
401
402Self-check passes...
403x ^ 3 + x + 2, 3 is NOT primitive!
404
405 - A Program for Computing Primitive Polynomials.
406
407Primpoly comes with ABSOLUTELY NO WARRANTY; for details see the
408GNU General Public License. This is free software, and you are welcome
409to redistribute it under certain conditions; see the GNU General Public License
410for details.
411
412Self-check passes...
413x ^ 2 + x + 10, 32749 is primitive!