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Ten new primitive binary trinomials

Author(s): Richard P. Brent; Paul Zimmermann.
Journal: Math. Comp.
MSC (2000): Primary 11B83, 11Y16; Secondary 11-04, 11T06, 11Y55, 12-04
Posted: August 1, 2008
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Abstract: We exhibit ten new primitive trinomials over GF(2) of record degrees $ 24\,036\,583$, $ 25\,964\,951$, $ 30\,402\,457$, and $ 32\,582\,657$. This completes the search for the currently known Mersenne prime exponents.

References:

1.
Richard P. Brent, Search for primitive trinomials (mod 2), http://wwwmaths.anu.edu.au/~brent/trinom.html, 2008.

2.
Richard Brent, Pierrick Gaudry, Emmanuel Thomé, and Paul Zimmermann, Faster multiplication in $ {\rm GF}(2)[x]$, Proc. of the 8th International Symposium on Algorithmic Number Theory (ANTS VIII), Lecture Notes in Computer Science 5011, Springer-Verlag, 2008, 153-166.

3.
Richard P. Brent, Samuli Larvala, and Paul Zimmermann, A fast algorithm for testing reducibility of trinomials mod $ 2$ and some new primitive trinomials of degree $ 3021377$, Math. Comp. 72 (2003), 1443-1452. MR 1972745 (2004b:11161)

4.
-, A primitive trinomial of degree $ 6972593$, Math. Comp. 74 (2005), 1001-1002. MR 2114660 (2005h:11054)

5.
Richard P. Brent and Paul Zimmermann, A multi-level blocking distinct degree factorization algorithm (extended abstract), Proceedings of the 8th International Conference on Finite Fields and Applications (Fq8) (Melbourne, Australia), 2007.

6.
-, A multi-level blocking distinct degree factorization algorithm, Contemporary Mathematics, special issue, to appear. Also available as

arXiv:0710.4410.

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The Great Internet Mersenne Prime Search, http://mersenne.org/.

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J. R. Heringa, H. W. J. Blöte, and A. Compagner, New primitive trinomials of Mersenne-exponent degrees for random-number generation, International Journal of Modern Physics C 3 (1992), 561-564. MR 1169571 (94a:11118)

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T. Kumada, H. Leeb, Y. Kurita, and M. Matsumoto, New primitive $ t$-nomials $ (t = 3$, $ 5)$ over $ {\rm GF}(2)$ whose degree is a Mersenne exponent, Math. Comp. 69 (2000), 811-814; MR 1665959 (2000i:11183); corrigenda: ibid 71 (2002), 1337-1338; MR 1898761 (2003c:11153)

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Y. Kurita and M. Matsumoto, Primitive $ t$-nomials $ (t = 3, 5)$ over $ {\rm GF}(2)$ whose degree is a Mersenne exponent $ \le 44497$, Math. Comp. 56 (1991), 817-821. MR 1068813 (91h:11138)

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Victor Shoup, NTL: A library for doing number theory, http://www.shoup.net/ntl/, 2007.

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Additional Information:

Richard P. Brent
Affiliation: Australian National University, Canberra, Australia
Email: trinomials@rpbrent.com

Paul Zimmermann
Affiliation: INRIA Nancy, Grand Est, Villers-lès-Nancy, France
Email: Paul.Zimmermann@loria.fr

DOI: 10.1090/S0025-5718-08-02170-4
PII: S 0025-5718(08)02170-4
Keywords: $ {GF}(2)[x]$, irreducible polynomials, irreducible trinomials, primitive polynomials, primitive trinomials, Mersenne exponents, Mersenne numbers
Received by editor(s): April 15, 2008
Posted: August 1, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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