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Old 2007-09-06, 20:23   #1
akruppa's Avatar
Aug 2002

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Default New primitive trinomials

Paul Zimmermann posted this to NMBRTHRY today:

we are proud to announce two primitive trinomials of huge degree over GF(2):

x24036583 + x8412642 + 1
x24036583 + x8785528 + 1

We also provide a certificate of reducibility for all other trinomials of
degree 24036583 (the 41st Mersenne exponent found), which can be checked in
less than ten hours with Magma [1]. This proves that the above two trinomials
(and the reverse trinomials) are the only primitive trinomials of degree
24036583 over GF(2). We thank Allan Steel, who did check both trinomials are
irreducible with Magma.

Unlike previous search [2], we used a new algorithm, outlined in [3], which
was presented at the Fq8 conference last July. This new algorithm both improves
the asymptotic complexity of the search for primitive trinomials, and also
provides certificates of divisibility for non-irreducible trinomials.

Richard Brent
Paul Zimmermann

[2] Richard P. Brent, Samuli Larvala, Paul Zimmermann, A Primitive Trinomial
of Degree 6972593, Math. of Comp., 74 (250), 2005.
[3] A Multi-level Blocking Distinct Degree Factorization Algorithm,
Richard P. Brent, Paul Zimmermann, 8th International Conference on Finite
Fields and Applications (Fq8), July 2007.
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