1582:
886:
1577:{\displaystyle {\begin{aligned}x^{3}+2x+1&=(x-\gamma )(x-\gamma ^{3})(x-\gamma ^{9})\\x^{3}+2x^{2}+x+1&=(x-\gamma ^{5})(x-\gamma ^{5\cdot 3})(x-\gamma ^{5\cdot 9})=(x-\gamma ^{5})(x-\gamma ^{15})(x-\gamma ^{19})\\x^{3}+x^{2}+2x+1&=(x-\gamma ^{7})(x-\gamma ^{7\cdot 3})(x-\gamma ^{7\cdot 9})=(x-\gamma ^{7})(x-\gamma ^{21})(x-\gamma ^{11})\\x^{3}+2x^{2}+1&=(x-\gamma ^{17})(x-\gamma ^{17\cdot 3})(x-\gamma ^{17\cdot 9})=(x-\gamma ^{17})(x-\gamma ^{25})(x-\gamma ^{23}).\end{aligned}}}
34:
1752:
307:
1870:. Primitive polynomials, or multiples of them, are sometimes a good choice for generator polynomials because they can reliably detect two bit errors that occur far apart in the message bitstring, up to a distance of
1866:(CRC) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF(2) and dividing it by a fixed generator polynomial also over GF(2); see
891:
51:
1631:
1818:
1782:
373:
is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by
219:
2018:
98:
70:
77:
20:
2051:
143:
117:
84:
517:
147:
2131:
66:
1898:
1832:
55:
1828:
2187:
870:. The other primitive polynomials are associated with algebraically conjugate sets built on other primitive elements
510:
839:
primitive elements. As each primitive polynomial of degree 3 has three roots, all necessarily primitive, there are
131:
1935:
91:
1886:
A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms:
1863:
324:
44:
629:
384:
352:
1761:
of the nonzero elements of the finite field, by representing an element by the corresponding exponent of
1747:{\displaystyle \mathrm {GF} (p^{m})=\{0,1=\alpha ^{0},\alpha ,\alpha ^{2},\ldots ,\alpha ^{p^{m}-2}\}.}
358:
A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by
2192:
456:
2135:
1867:
1785:
1843:
is the length of the linear-feedback shift register) may be built from a primitive polynomial.
2160:
2057:
2047:
1790:
1764:
1901:. A number of results give techniques for locating and testing primitiveness of trinomials.
2127:
2102:
2084:
1929:
199:
1784:
This representation makes multiplication easy, as it corresponds to addition of exponents
2163:
2030:
C. Paar, J. Pelzl - Understanding
Cryptography: A Textbook for Students and Practitioners
1932:
pseudo-random number generator does not use a trinomial, it does take advantage of this.
1916:
is primitive if and only if it is irreducible. (Given an irreducible polynomial, it is
1909:
2088:
2181:
2106:
628:
follows from the property that the polynomial is invariant under application of the
19:
For polynomials such that the greatest common divisor of the coefficients is 1, see
2075:
Zierler, Neal; Brillhart, John (December 1968). "On primitive trinomials (Mod 2)".
1598:
466:
151:
677:: its roots generate a cyclic group of order 4, while the multiplicative group of
1949:. This can be used to create a pseudo-random number generator of the huge period
302:{\displaystyle \{0,1,\alpha ,\alpha ^{2},\alpha ^{3},\ldots \alpha ^{p^{m}-2}\}}
135:
33:
1827:
Primitive polynomials over GF(2), the field with two elements, can be used for
794:
are also algebraically conjugate and produce the second primitive polynomial:
2168:
2061:
642:) and from the fact that the fixed field of the Frobenius automorphism is
2041:
1758:
696:. Then, because the natural numbers less than and relatively prime to
406:
is prime, is a primitive polynomial if the smallest positive integer
2140:
363:
1597:
Primitive polynomials can be used to represent the elements of a
1938:
has been tabulating primitive trinomials of this form, such as
843:
primitive polynomials of degree 3. One primitive polynomial is
692:, on the other hand, is primitive. Denote one of its roots by
608:. That the coefficients of a polynomial of this form, for any
27:
2014:
2010:
2006:
2002:
1998:
16:
Minimal polynomial of a primitive element in a finite field
1897:. Their simplicity makes for particularly small and fast
2134:(24 May 2016). "Twelve new primitive binary trinomials".
1854:
pseudo-random bits before repeating the same sequence.
1928:. Primes have no non-trivial factors.) Although the
1977:
Enumerations of primitive polynomials by degree over
1793:
1767:
1634:
889:
222:
670:
is irreducible but not primitive because it divides
58:. Unsourced material may be challenged and removed.
1812:
1776:
1746:
1576:
301:
1846:In general, for a primitive polynomial of degree
355:, all primitive polynomials are also irreducible.
2043:Random number generation and Monte Carlo methods
700:are 1, 3, 5, and 7, the four primitive roots in
2107:"Search for Primitive Trinomials (mod 2)"
2046:(2 ed.). New York: Springer. p. 39.
1757:This allows an economical representation in a
681:is a cyclic group of order 8. The polynomial
8:
1738:
1662:
296:
223:
858:, the algebraically conjugate elements are
1621:) are represented as successive powers of
2139:
1798:
1792:
1766:
1724:
1719:
1700:
1681:
1650:
1635:
1633:
1558:
1536:
1514:
1483:
1455:
1433:
1401:
1385:
1368:
1346:
1324:
1293:
1265:
1243:
1202:
1189:
1172:
1150:
1128:
1097:
1069:
1047:
1009:
993:
976:
954:
898:
890:
888:
465:, meaning that any of them generates the
282:
277:
261:
248:
221:
118:Learn how and when to remove this message
2019:Online Encyclopedia of Integer Sequences
67:"Primitive polynomial" field theory
1970:
1850:over GF(2), this process will generate
1609:) is a root of a primitive polynomial
501:primitive polynomials, each of degree
7:
1835:with maximum cycle length (which is
750:are algebraically conjugate. Indeed
620:, not necessarily primitive, lie in
351:Because all minimal polynomials are
56:adding citations to reliable sources
1617:), then the nonzero elements of GF(
1904:For polynomials over GF(2), where
1639:
1636:
21:Primitive polynomial (ring theory)
14:
439:A primitive polynomial of degree
1920:primitive only if the period of
777:. The remaining primitive roots
554:and so the primitive polynomial
32:
1899:linear-feedback shift registers
854:. Denoting one of its roots by
161:. This means that a polynomial
43:needs additional citations for
1833:linear-feedback shift register
1656:
1643:
1564:
1545:
1542:
1523:
1520:
1501:
1495:
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1442:
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1355:
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1305:
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1178:
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1034:
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963:
960:
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926:
1:
2089:10.1016/S0019-9958(68)90973-X
1823:Pseudo-random bit generation
1593:Field element representation
1924:is a non-trivial factor of
1829:pseudorandom bit generation
632:to its coefficients (using
2209:
18:
2040:Gentle, James E. (2003).
1912:, a polynomial of degree
1813:{\displaystyle p^{m}-1.}
1777:{\displaystyle \alpha .}
880:relatively prime to 26:
511:Euler's totient function
2077:Information and Control
1997:are given by sequences
1864:cyclic redundancy check
520:of a primitive element
487:primitive elements and
2164:"Primitive Polynomial"
1878:primitive polynomial.
1814:
1778:
1748:
1578:
740:. The primitive roots
630:Frobenius automorphism
385:irreducible polynomial
303:
1815:
1779:
1749:
1579:
380:(it has 1 as a root).
304:
176:with coefficients in
2083:(6): 541, 548, 553.
1882:Primitive trinomials
1791:
1765:
1632:
887:
518:algebraic conjugates
476:) there are exactly
467:multiplicative group
317:. This implies that
309:is the entire field
220:
196:primitive polynomial
140:primitive polynomial
52:improve this article
2188:Field (mathematics)
447:different roots in
132:finite field theory
2161:Weisstein, Eric W.
1868:Mathematics of CRC
1810:
1774:
1744:
1574:
1572:
565:has explicit form
299:
144:minimal polynomial
2128:Brent, Richard P.
2103:Brent, Richard P.
1831:. In fact, every
455:, which all have
148:primitive element
128:
127:
120:
102:
2200:
2174:
2173:
2146:
2145:
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2132:Zimmermann, Paul
2124:
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2113:
2105:(4 April 2016).
2099:
2093:
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2037:
2031:
2028:
2022:
1996:
1992:
1988:
1984:
1980:
1975:
1960:
1958:
1952:
1948:
1930:Mersenne Twister
1927:
1907:
1896:
1873:
1853:
1838:
1819:
1817:
1816:
1811:
1803:
1802:
1783:
1781:
1780:
1775:
1753:
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1750:
1745:
1737:
1736:
1729:
1728:
1705:
1704:
1686:
1685:
1655:
1654:
1642:
1583:
1581:
1580:
1575:
1573:
1563:
1562:
1541:
1540:
1519:
1518:
1494:
1493:
1466:
1465:
1438:
1437:
1406:
1405:
1390:
1389:
1373:
1372:
1351:
1350:
1329:
1328:
1304:
1303:
1276:
1275:
1248:
1247:
1207:
1206:
1194:
1193:
1177:
1176:
1155:
1154:
1133:
1132:
1108:
1107:
1080:
1079:
1052:
1051:
1014:
1013:
998:
997:
981:
980:
959:
958:
903:
902:
879:
875:
869:
863:
857:
853:
842:
838:
827:
820:
793:
782:
776:
749:
743:
739:
728:
718:
707:
703:
699:
695:
691:
680:
676:
669:
662:
649:
641:
627:
619:
611:
607:
564:
553:
547:
541:
535:
531:
523:
508:
504:
500:
486:
464:
454:
446:
442:
435:
424:
379:
372:
341:
331:
322:
316:
308:
306:
305:
300:
295:
294:
287:
286:
266:
265:
253:
252:
215:
207:
193:
175:
171:
160:
123:
116:
112:
109:
103:
101:
60:
36:
28:
2208:
2207:
2203:
2202:
2201:
2199:
2198:
2197:
2178:
2177:
2159:
2158:
2155:
2150:
2149:
2126:
2125:
2121:
2111:
2109:
2101:
2100:
2096:
2074:
2073:
2069:
2054:
2039:
2038:
2034:
2029:
2025:
1994:
1990:
1986:
1982:
1978:
1976:
1972:
1967:
1956:
1954:
1950:
1939:
1925:
1905:
1887:
1884:
1871:
1860:
1851:
1836:
1825:
1794:
1789:
1788:
1763:
1762:
1720:
1715:
1696:
1677:
1646:
1630:
1629:
1595:
1590:
1571:
1570:
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1510:
1479:
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1429:
1413:
1397:
1381:
1378:
1377:
1364:
1342:
1320:
1289:
1261:
1239:
1223:
1198:
1185:
1182:
1181:
1168:
1146:
1124:
1093:
1065:
1043:
1027:
1005:
989:
986:
985:
972:
950:
919:
894:
885:
884:
877:
871:
865:
859:
855:
844:
840:
829:
825:
795:
784:
778:
751:
745:
741:
730:
720:
709:
705:
701:
697:
693:
682:
678:
671:
664:
663:the polynomial
660:
657:
643:
633:
621:
613:
609:
566:
555:
549:
543:
537:
533:
525:
521:
506:
502:
488:
477:
459:
448:
444:
440:
426:
419:
374:
367:
348:
335:
332:)-root of unity
326:
318:
310:
278:
273:
257:
244:
218:
217:
209:
203:
202:and has a root
177:
173:
162:
154:
124:
113:
107:
104:
61:
59:
49:
37:
24:
17:
12:
11:
5:
2206:
2204:
2196:
2195:
2190:
2180:
2179:
2176:
2175:
2154:
2153:External links
2151:
2148:
2147:
2119:
2094:
2067:
2052:
2032:
2023:
1969:
1968:
1966:
1963:
1910:Mersenne prime
1883:
1880:
1859:
1856:
1824:
1821:
1809:
1806:
1801:
1797:
1773:
1770:
1755:
1754:
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1670:
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1664:
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1658:
1653:
1649:
1645:
1641:
1638:
1594:
1591:
1589:
1586:
1585:
1584:
1569:
1566:
1561:
1557:
1553:
1550:
1547:
1544:
1539:
1535:
1531:
1528:
1525:
1522:
1517:
1513:
1509:
1506:
1503:
1500:
1497:
1492:
1489:
1486:
1482:
1478:
1475:
1472:
1469:
1464:
1461:
1458:
1454:
1450:
1447:
1444:
1441:
1436:
1432:
1428:
1425:
1422:
1419:
1416:
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1412:
1409:
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1400:
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1393:
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1268:
1264:
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1254:
1251:
1246:
1242:
1238:
1235:
1232:
1229:
1226:
1224:
1222:
1219:
1216:
1213:
1210:
1205:
1201:
1197:
1192:
1188:
1184:
1183:
1180:
1175:
1171:
1167:
1164:
1161:
1158:
1153:
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1145:
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1136:
1131:
1127:
1123:
1120:
1117:
1114:
1111:
1106:
1103:
1100:
1096:
1092:
1089:
1086:
1083:
1078:
1075:
1072:
1068:
1064:
1061:
1058:
1055:
1050:
1046:
1042:
1039:
1036:
1033:
1030:
1028:
1026:
1023:
1020:
1017:
1012:
1008:
1004:
1001:
996:
992:
988:
987:
984:
979:
975:
971:
968:
965:
962:
957:
953:
949:
946:
943:
940:
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934:
931:
928:
925:
922:
920:
918:
915:
912:
909:
906:
901:
897:
893:
892:
824:For degree 3,
656:
653:
652:
651:
514:
470:
437:
381:
356:
347:
344:
298:
293:
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285:
281:
276:
272:
269:
264:
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237:
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134:, a branch of
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38:
31:
15:
13:
10:
9:
6:
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3:
2:
2205:
2194:
2191:
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2183:
2171:
2170:
2165:
2162:
2157:
2156:
2152:
2142:
2137:
2133:
2129:
2123:
2120:
2108:
2104:
2098:
2095:
2090:
2086:
2082:
2078:
2071:
2068:
2063:
2059:
2055:
2053:0-387-00178-6
2049:
2045:
2044:
2036:
2033:
2027:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1974:
1971:
1964:
1962:
1946:
1942:
1937:
1936:Richard Brent
1933:
1931:
1923:
1919:
1915:
1911:
1902:
1900:
1894:
1890:
1881:
1879:
1877:
1874:for a degree
1869:
1865:
1857:
1855:
1849:
1844:
1842:
1834:
1830:
1822:
1820:
1807:
1804:
1799:
1795:
1787:
1771:
1768:
1760:
1741:
1733:
1730:
1725:
1721:
1716:
1712:
1709:
1706:
1701:
1697:
1693:
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1687:
1682:
1678:
1674:
1671:
1668:
1665:
1659:
1651:
1647:
1628:
1627:
1626:
1624:
1620:
1616:
1612:
1608:
1604:
1600:
1592:
1587:
1567:
1559:
1555:
1551:
1548:
1537:
1533:
1529:
1526:
1515:
1511:
1507:
1504:
1498:
1490:
1487:
1484:
1480:
1476:
1473:
1462:
1459:
1456:
1452:
1448:
1445:
1434:
1430:
1426:
1423:
1417:
1415:
1410:
1407:
1402:
1398:
1394:
1391:
1386:
1382:
1369:
1365:
1361:
1358:
1347:
1343:
1339:
1336:
1325:
1321:
1317:
1314:
1308:
1300:
1297:
1294:
1290:
1286:
1283:
1272:
1269:
1266:
1262:
1258:
1255:
1244:
1240:
1236:
1233:
1227:
1225:
1220:
1217:
1214:
1211:
1208:
1203:
1199:
1195:
1190:
1186:
1173:
1169:
1165:
1162:
1151:
1147:
1143:
1140:
1129:
1125:
1121:
1118:
1112:
1104:
1101:
1098:
1094:
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1087:
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1066:
1062:
1059:
1048:
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1040:
1037:
1031:
1029:
1024:
1021:
1018:
1015:
1010:
1006:
1002:
999:
994:
990:
977:
973:
969:
966:
955:
951:
947:
944:
935:
932:
929:
923:
921:
916:
913:
910:
907:
904:
899:
895:
883:
882:
881:
874:
868:
862:
851:
847:
836:
832:
822:
818:
814:
810:
806:
802:
798:
791:
787:
781:
774:
770:
766:
762:
758:
754:
748:
737:
733:
727:
723:
716:
712:
689:
685:
674:
667:
654:
647:
640:
636:
631:
625:
617:
605:
601:
597:
593:
589:
585:
581:
577:
573:
569:
562:
558:
552:
546:
540:
529:
519:
515:
512:
499:
495:
491:
484:
480:
475:
471:
469:of the field.
468:
462:
458:
452:
438:
433:
429:
422:
417:
413:
409:
405:
401:
397:
393:
389:
386:
382:
377:
370:
365:
361:
357:
354:
350:
349:
345:
343:
339:
333:
329:
321:
314:
291:
288:
283:
279:
274:
270:
267:
262:
258:
254:
249:
245:
241:
238:
235:
232:
229:
226:
213:
206:
201:
197:
192:
189:
185:
181:
169:
165:
158:
153:
149:
145:
141:
137:
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1599:finite field
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325:primitive (
136:mathematics
2182:Categories
2141:1605.09213
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410:such that
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