We give the results of a computer search for maximally equidistributed combined linear feedback shift register (or Tausworthe) random number generators, whose components are trinomials of degrees slightly less than 32 or 64. These generators are fast and have good statistical properties.

Let q be a power of 2. Recently, Tu and Zeng considered trinomials the form f ( X ) = + 1 ? 4 2 ? b 3 , where ? F ? . They proved that is permutation polynomial if has no root in In this paper, we show above sufficient condition also necessary.

All primitive trinomials over GF (2) with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are X859433 +X288477 + 1 and its reciprocal. Also two examples of primitive pentanomials over GF (2) with degree 86243 (which is the 28th Mersenne exponent) are presented. The sieve used is briefly described.

We construct classes of permutation polynomials over FQ2 by exhibiting classes of low-degree rational functions over FQ2 which induce bijections on the set of (Q + 1)-th roots of unity. As a consequence, we prove two conjectures about permutation trinomials from a recent paper by Tu, Zeng, Hu and Li.