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The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r +O(1) bits of memory. We describe a new algorithm which requires only 3r=2+O(1) bits of memory and signicantly fewer memory references and bit-operations than the standard algorithm. If 2r 1 is a Mersenne prime, then an irreducible trinomial of degree r is necessarily primitive. We give primitive trinomials for the Mersenne exponents r = 756839, 859433, and 3021377. The results for r = 859433 extend and correct some computations of Kumada et al. The two results for r = 3021377 are primitive trinomials of the highest known degree. Throughout this paper all polynomials are assumed to be in Z2(x). A polynomial P (x )i sreducible if it has nontrivial factors; otherwise it is irreducible .A polynomial P (x )o f degree r> 1i sprimitive if P (x) is irreducible and xj 6 =1 modP (x )f or 0 <j< 2r 1. If P (x )i s primitive, thenx is a generator for the multiplicative group of the eld Z2(x)=(P (x)), giving a useful representation of GF(2r). See Lidl

A fast algorithm for testing reducibility of trinomials mod~2 and some new primitive trinomials of degree 3021377