Irreducible trinomials of given degree n over F2 do not always exist and in the cases that there is no irreducible trinomial of degree n it may be effective to use trinomials with an irreducible factor of degree n. In this paper we consider some conditions under which irreducible polynomials divide trinomials over F2. A condition for divisibility of selfreciprocal trinomials by irreducible poly...

Consider polynomials over GF(2). We describe efficient algorithms for finding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree r for all Mersenne exponents r = ±3 mod 8 in the range 5 < r < 10, although there is no irreducible trinomial of degree r. We also give trinomials with a primitive factor of degree r ...

Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six new classes of permutation trinomials over finite fields of even characteristic are constructed from six fractional polynomials. Further, three classes of p...

The construction of permutation trinomials over finite fields attracts people’s interest recently due to their simple form and some additional properties. Motivated by some results on the construction of permutation trinomials with Niho exponents, by constructing some new fractional polynomials that permute the set of the (q + 1)-th roots of unity in Fq2 , we present several classes of permutat...

In this paper, we give conditions under which the trinomials of the form x + ax + b over finite field Fpm are not primitive and conditions under which there are no primitive trinomials of the form x +ax+b over finite field Fpm . For finite field F4, We show that there are no primitive trinomials of the form x + x + α, if n ≡ 1 mod 3 or n ≡ 0 mod 3 or n ≡ 4 mod 5.

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, ...

In this paper, we explore the primitivity of trinomials over small finite fields. We extend the results of the primitivity of trinomials x + ax + b over F4 [1] to the general form x + ax + b. We prove that for given n and k, one of all the trinomials x + ax + b with b being the primitive element of F4 and a + b 6= 1 is primitive over F4 if and only if all the others are primitive over F4. And w...

We exhibit ten new primitive trinomials over GF(2) of record degrees 24 036 583, 25 964 951, 30 402 457, and 32 582 657. This completes the search for the currently known Mersenne prime exponents. Primitive trinomials of degree up to 6 972 593 were previously known [4]. We have completed a search for all known Mersenne prime exponents [7]. Ten new primitive trinomials were found. Our results ar...

We exhibit twelve new primitive trinomials over GF(2) of record degrees 42 643 801, 43 112 609, and 74 207 281. In addition we report the first Mersenne exponent not ruled out by Swan’s theorem [10] — namely 57 885 161 — for which none primitive trinomial exists. This completes the search for the currently known Mersenne prime exponents. Primitive trinomials of degree up to 32 582 657 were repo...