Ten new primitive binary trinomials

Abstract

Abstract. 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 are summarized in the following theorem: Theorem 1. For the integers r listed in Table 1, the primitive trinomials x r +x s +1 of degree r over GF(2) are exactly those given in Table 1, and the corresponding reciprocal trinomials x r + x r−s + 1. Proof. From the GIMPS Project [7], the integers r listed in Table 1 are exponents of Mersenne primes 2 r − 1. Thus, irreducible trinomials of degree r are necessarily primitive. Irreducibility of the trinomials listed in Table 1 follows from the authors' computations, using the new algorithm described in [5,6] (verified using the algorithm of [3] and independently verified by Allan Steel using Magma). Finally, the fact that no irreducible trinomials were missed during the search, for those degrees r, follows from the certificates given on the authors' web pages [1]. Table 1 are the known Mersenne exponents of the form r = ±1 mod 8 in the interval [100 000, 32 582 657]. For smaller exponents, omitted to save space, see [10] or our web site [1]. According to the GIMPS Project [7], the list is complete for r ≤ 16 300 000. Known Mersenne exponents of the form r = ±3 mod 8 for r > 5 cannot be the degrees of irreducible trinomials due to Swan's theorem [12]; the possibility x r +x 2 +1 permitted by Swan's theorem is easily ruled out in all known cases with r > 5; see the authors' web site [1]. Remarks. The integers r listed inOur search used a new algorithm [5,6] relying on fast arithmetic in GF(2) [x], whose details are given in [2]. Another significant improvement over previous work is that certificates were produced; this enables one to check easily that the claimed nonprimitive trinomials are indeed reducible. A certificate is simply an encoding of a nontrivial factor of smallest degree. A 2.4Ghz Intel Core 2 takes only 15 minutes to check the certificates of all 16 291 325 reducible trinomials (s ≤ r/2) of degree r = 32 582 657 with our check-ntl program based on NTL [11].

The great trinomial hunt

“…, r ′ 44 by January 2008. Once again, we thought we were finished and published our results [7] If the GCD has degree λd for λ > 1, and one wants to split the product into λ factors of degree d, then an equal degree factorization algorithm (EDF) is used. If the EDF is necessary it is usually cheap, since the total degree λd is usually small if λ > 1.…”

The MIXMAX random number generator

“…However, the required amount of skipping would likely be prohibitively expensive. Moreover, it is clear that the recurrences based on primitive trinomials of even higher order such as those found in [17] would exhibit even longer correlations.…”

Swan-like results for binomials and trinomials over finite fields of odd characteristic

“…As a corollary, Swan proves that there is no irreducible trinomial over F 2 of degree 8k, where k is any positive integer [13]. Swan's theorem has recently been used by Brent and Zimmerman [3] to reduce the number of cases in a search for primitive trinomials over F 2 of enormous degree.…”