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:
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
51:
219:
98:
70:
77:
2018:
84:
66:
20:
2051:
870:. The other primitive polynomials are associated with algebraically conjugate sets built on other primitive elements
143:
117:
839:
primitive elements. As each primitive polynomial of degree 3 has three roots, all necessarily primitive, there are
517:
147:
2131:
1898:
1832:
91:
55:
1828:
1886:
A useful class of primitive polynomials is the primitive trinomials, those having only three nonzero terms:
2187:
1761:
of the nonzero elements of the finite field, by representing an element by the corresponding exponent of
510:
1747:{\displaystyle \mathrm {GF} (p^{m})=\{0,1=\alpha ^{0},\alpha ,\alpha ^{2},\ldots ,\alpha ^{p^{m}-2}\}.}
131:
358:
A primitive polynomial must have a non-zero constant term, for otherwise it will be divisible by
1935:
466:
1863:
324:
44:
1843:
is the length of the linear-feedback shift register) may be built from a primitive polynomial.
629:
384:
352:
1790:
1764:
1901:. A number of results give techniques for locating and testing primitiveness of trinomials.
2192:
456:
1784:
This representation makes multiplication easy, as it corresponds to addition of exponents
8:
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
2135:
1867:
1785:
2088:
2160:
2106:
2057:
2047:
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
2127:
2102:
2084:
2075:
Zierler, Neal; Brillhart, John (December 1968). "On primitive trinomials (Mod 2)".
1929:
199:
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}\}}
1909:
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:
2181:
2061:
1598:
151:
642:) and from the fact that the fixed field of the Frobenius automorphism is
2041:
135:
696:. Then, because the natural numbers less than and relatively prime to
2168:
406:
is prime, is a primitive polynomial if the smallest positive integer
33:
2140:
1758:
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
2158:
363:
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.
2179:
2074:
2043:Random number generation and Monte Carlo methods
700:are 1, 3, 5, and 7, the four primitive roots in
2126:
1822:
1592:
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
1738:
1662:
296:
223:
858:, the algebraically conjugate elements are
1621:) are represented as successive powers of
2139:
465:, meaning that any of them generates the
118:Learn how and when to remove this message
2019:Online Encyclopedia of Integer Sequences
67:"Primitive polynomial" field theory
1881:
1850:over GF(2), this process will generate
2180:
2039:
1609:) is a root of a primitive polynomial
501:primitive polynomials, each of degree
2159:
2101:
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
27:
1617:), then the nonzero elements of GF(
13:
1904:For polynomials over GF(2), where
1639:
1636:
21:Primitive polynomial (ring theory)
14:
2204:
2152:
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
1587:
854:. Denoting one of its roots by
161:. This means that a polynomial
43:needs additional citations for
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2095:
2068:
2033:
2024:
1971:
1833:linear-feedback shift register
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1:
2089:10.1016/S0019-9958(68)90973-X
1964:
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1857:
1823:Pseudo-random bit generation
1593:Field element representation
7:
1924:is a non-trivial factor of
1829:pseudorandom bit generation
654:
632:to its coefficients (using
10:
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:
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380:(it has 1 as a root).
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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:
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565:has explicit form
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144:minimal polynomial
2128:Brent, Richard P.
2103:Brent, Richard P.
1831:. In fact, every
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148:primitive element
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2132:Zimmermann, Paul
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2105:(4 April 2016).
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751:
745:
741:
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697:
693:
682:
678:
671:
664:
663:the polynomial
660:
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643:
633:
621:
613:
609:
566:
555:
549:
543:
537:
533:
525:
521:
506:
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488:
477:
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448:
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440:
426:
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374:
367:
348:
335:
332:)-root of unity
326:
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277:
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261:
257:
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244:
221:
218:
217:
209:
203:
202:and has a root
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113:
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104:
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59:
49:
37:
24:
17:
12:
11:
5:
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2153:External links
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1969:
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1910:Mersenne prime
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824:For degree 3,
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2053:0-387-00178-6
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1874:for a degree
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469:of the field.
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325:primitive (
136:mathematics
2182:Categories
2141:1605.09213
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