# Irreducible polynomial on $\mathbb{F}_{2}[x]$

For some reason I need some irreducible polynomial $f$ on $\mathbb{F}_{2}[x]$ where $\deg f \in [10^3,10^6]$. Could someone give information about this? Thx.

• Are you just asking for one example? – Jason Starr Apr 11 '14 at 11:37
• $x^{1001}+x^{17}+1$ – Felipe Voloch Apr 11 '14 at 12:21
• To Jason Starr, the more, the better. – Lwins Apr 12 '14 at 0:14

You might take a look at the paper "A fast algorithm for testing reducibility of trinomials mod 2 and some new primitive trinomials of degree 3021377'' by Richard P. Brent, Samuli Larvala and Paul Zimmerman published in Mathematics of Computation in 2003 (pages 1443-1452) (I think the article is free online here). They give some specific examples of irreducible polynomials over $\mathbb{F}_{2}[x]$ of the form $x^{p} + x^{s} + 1$ for several prime numbers $p$ between $10^{5}$ and $10^{6}$. This is at the top end of the range you're looking for - they also have some references to older papers with smaller degree polynomials.