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Generators and Primitive Trinomials Ri hard P. Brent Computing Laboratory University of Oxford rpb omlab.ox.a .uk 1 May 2001 To be presented at OUCL, 1 May 2001. Copyright 2001, R. P. Brent. oxford3t Abstra t In this talk, whi h des ribes joint work with Samuli Larvala and Paul Zimmermann, we onsider the problem of testing trinomials over GF(2) for irredu ibility or primitivity. In parti ular, we onsider trinomials whose degree is the exponent of a Mersenne prime. We des ribe a new algorithm for testing su h trinomials. The algorithm is signi antly faster than the standard algorithm, and has been used to nd primitive trinomials of degree 3021377. Previously, the highest degree known was 859433. One of the appli ations is to pseudo-random number generators. Using the new primitive trinomials, we an obtain uniform random number generators with extremely long period and good statisti al properties in all dimensions less than 3021377. 2 Outline De nitions Conne tion with RNGs Sieving The standard algorithm The new algorithm Performan e Some new primitive trinomials Appli ation to RNGs