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!