Primitive t-Nomials (t = 3, 5) Over GF(2) Whose Degree is a Mersenne Exponent 44497

Kurita, Yoshiharu; Matsumoto, Makoto

doi:10.2307/2008411

Cited by 12 publications

(17 citation statements)

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“…Although complete descriptions are not always given in the original papers, previous computations [9,10,12,14,20,21,24,26] involving irreducible or primitive trinomials seem to have used some variant of sieving combined with squaring and reduction, much as described above. This is why we call it the standard algorithm.…”

Section: Reduction Mod P (X) Reduction Of A(x)mentioning

confidence: 99%

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

Brent

Larvala²,

Zimmermann

2002

Math. Comp.

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Abstract. 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 significantly fewer memory references and bit-operations than the standard algorithm.If 2 r − 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.

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Section: Reduction Mod P (X) Reduction Of A(x)mentioning

confidence: 99%

“…Kurita and Matsumoto [14] extended the search to r ≤ 86243, and Heringa et al [10] to r ≤ 216091. Kumada et al [12] conducted an exhaustive search for r = 859433 and found one primitive trinomial.…”

Section: Introductionmentioning

confidence: 99%

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

Brent

Larvala²,

Zimmermann

2002

Math. Comp.

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“…Some of them are: to use pentanomials [Kurita and Matsumoto 1991], to combine trinomials [Tezuka and L'Ecuyer 1991;Wang and Compagner 1993], and to twist [Matsumoto and Kurita 1994]. It seems that, for these generators also, the behavior around the fixed vector would be worth testing.…”

Section: Concluding Discussionmentioning

confidence: 99%

“…The most common way is to use three-term linear recursion, in other words, to use primitive trinomials as characteristic polynomial. These primitive trinomials are intensively searched in Heringa et al [1992] and Kurita and Matsumoto [1991]. In this paper, however, we reveal a serious flaw of trinomial generators.…”

Section: Characteristic M-sequencementioning

confidence: 93%

Strong deviations from randomness in m -sequences based on trinomials

Matsumoto

Kurita²

1996

ACM Trans. Model. Comput. Simul.

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“…The integers r listed in 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.…”

Section: Theorem 1 For the Integers R Listed Inmentioning

confidence: 99%

Ten new primitive binary trinomials

Brent

Zimmermann

2008

Math. Comp.

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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].

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Primitive t-Nomials (t = 3, 5) Over GF(2) Whose Degree is a Mersenne Exponent 44497

Cited by 12 publications

References 0 publications

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

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

Strong deviations from randomness in m -sequences based on trinomials

Ten new primitive binary trinomials

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