Primitive Trinomials Research Articles

Ten new primitive binary trinomials

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.

Some long-period random number generators using shifts and xors

Marsaglia recently introduced a class of `xorshift' random number generators with periods \(2^n-1\) for \(n = 32, 64,\ldots\). Here Marsaglia's xorshift generators are generalised to obtain fast and high quality random number generators with extremely long periods. Whereas random number generators based on primitive trinomials may be unsatisfactory, because a trinomial has very small weight, these new generators can be chosen so that their minimal polynomials have a large number of non-zero terms and, hence, a large weight. A computer search using Magma found good random number generators for\(~n\) a power of two up to 4096. These random number generators are implemented in a free software package \texttt{xorgens}.

The nonlinear complexity of level sequences over [formula omitted

For any sequence a ̲ over Z / ( 2 2 ) , there is an unique 2-adic expansion a ̲ = a ̲ 0 + a ̲ 1 · 2 , where a ̲ 0 and a ̲ 1 are sequences over { 0 , 1 } and can be regarded as sequences over the binary field GF ( 2 ) naturally. We call a ̲ 0 and a ̲ 1 the level sequences of a ̲ . Let f ( x ) be a primitive polynomial of degree n over Z / ( 2 2 ) , and a ̲ be a primitive sequence generated by f ( x ) . In this paper, we discuss how many bits of a ̲ 1 can determine uniquely the original primitive sequence a ̲ . This issue is equivalent with one to estimate the whole nonlinear complexity, NL ( f ( x ) , 2 2 ) , of all level sequences of f ( x ) . We prove that 4 n is a tight upper bound of NL ( f ( x ) , 2 2 ) if f ( x ) ( mod 2 ) is a primitive trinomial over GF ( 2 ) . Moreover, the experimental result shows that NL ( f ( x ) , 2 2 ) varies around 4 n if f ( x ) ( mod 2 ) is a primitive polynomial over GF ( 2 ) . From this result, we can deduce that NL ( f ( x ) , 2 2 ) is much smaller than L ( f ( x ) , 2 2 ) , where L ( f ( x ) , 2 2 ) is the linear complexity of level sequences of f ( x ) .

A primitive trinomial of degree 6972593

The only primitive trinomials of degree 6972593 6972593 over GF ⁡ ( 2 ) \operatorname {GF}(2) are x 6972593 + x 3037958 + 1 x^{6972593} + x^{3037958} + 1 and its reciprocal.

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

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 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.

Corrigenda to ``New primitive $t$-nomials $(t=3,5)$ over $GF(2)$ whose degree is a Mersenne exponent,'' and some new primitive pentanomials

We report an error in our previous paper [New primitive t-nomials $(t=3,5)$ over $GF(2)$ whose degree is a Mersenne exponent, Math. Comp. 69 (2000), no. 230, 811–814], where we announced that we listed all the primitive trinomials over $GF(2)$ of degree 859433, but there is a bug in the sieve. We missed the primitive trinomial $X^{859433}+X^{170340}+1$ and its reciprocal, as pointed out by Richard Brent et al. We also report some new primitive pentanomials.

New primitive $t$-nomials $(t = 3,5)$ over $GF(2)$ whose degree is a Mersenne exponent

All primitive trinomials over GF(2) with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are X 859433 + X 288477 + 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.

Orthogonal Arrays, Primitive Trinomials, and Shift-Register Sequences

We investigate properties of maximum-length binary shift-register sequences defined by primitive trinomials. We consider the set of subintervals of lengthnof such a sequence form<n≤2m+1, wheremis the degree of the trinomial. This set is an orthogonal array of strength 2 and is shown to have a property very close to being an orthogonal array of strength 3.

Four-tap shift-register-sequence random-number generators

Correlations in the generalized feedback shift-register random-number generator are shown to be greatly reduced when the number of feedback taps is increased from two to four (or more) and the tap offsets are made large. Simple formulas for producing maximal-cycle four-tap rules from available primitive trinomials are given, and explicit three- and four-point correlations are found for some of those rules. Several generators are also tested using a simple but sensitive random-walk simulation that relates to a problem in percolation theory. While virtually all two-tap generators fail this test, four-tap generators with offsets greater than about 500 pass it, have passed tests carried out by others, and appear to be good multipurpose high-quality random-number generators. © 1998 American Institute of Physics.

Maximally equidistributed combined Tausworthe generators

Tausworthe random number generators based on a primitive trinomial allow an easy and fast implementation when their parameters obey certain restrictions. However, such generators, with those restrictions, have bad statistical properties unless we combine them. A generator is called maximally equidistributed if its vectors of successive values have the best possible equidistribution in all dimensions. This paper shows how to find maximally equidistributed combinations in an efficient manner, and gives a list of generators with that property. Such generators have a strong theoretical support and lend themselves to very fast software implementations.

Table of Primitive Binary Polynomials. II

For those n < 5000, for which the factorization of 2n 1 is known, the first primitive trinomial (if such exists) and a randomly generated primitive 5and 7-nomial of degree n in GF(2) are given, if the respective entry is absent from the previously published table. After our paper with a list of primitive binary polynomials [3] had been accepted for publication, S. S. Wagstaff, Jr. made available to us reference [2] with some new factorizations of 2n 1 (updates to [1] obtained up to April 5, 1993). This enabled us to make an update to our list, in which the smallest degree of missing primitive binary polynomial is now 51 1. Here, like in our previous table, the degrees of nonzero coefficients (except 0 and n ) of primitive k-nomials (if there are any) of degree n are given for some missing 2 < n < 5000. TABLE 1. Primitive binary polynomials

A Table of Primitive Binary Polynomials

For those n < 5000, for which the factorization of 2n− 1 is known, the first primitive trinomial (if such exists) and a randomly generated primitive 5– and 7–nomial of degree n in GF(2) are given. A primitive polynomial of degree n over GF(2) is useful for generating a pseudo–random sequence of n–tuples of zeros and ones, see [8]. If the polynomial has a small number k of terms, then the sequence is easily computed. But for cryptological applications (correlation attack, see [5]) it is often necessary to have the primitive polynomials with k larger than one can find in the existing tables. For example, Zierler and Brillhart [10, 11] have calculated all irreducible trinomials of degree n ≤ 1000, with the period for some for which the factorization of 2n−1 is known; Stahnke [7] has listed one example of a trinomial or pentanomial of degree n ≤ 168; Zierler [12] has listed all primitive trinomials whose degree is a Mersenne exponent ≤ 11213 = M23 (here Mj denotes the jth Mersenne exponent); Rodemich and Rumsey [6] have listed all primitive trinomials of degree Mj, 12 ≤ j ≤ 17; Kurita and Matsumoto [2] have listed all primitive trinomials of degree Mj, 24 ≤ j ≤ 28, and one example of primitive pentanomials of degree Mj, 8 ≤ j ≤ 27. ∗1980 Mathematics Subject Classification (1985 Revision). Primary 11T06, 11T71. †

A table of primitive binary polynomials

For those n &gt; 5000 n &gt; 5000 for which the factorization of 2 n − 1 {2^n} - 1 is known, the first primitive trinomial (if such exists) and a randomly generated primitive 5- and 7-nomial of degree n in GF ( 2 ) {\text {GF}}(2) are given.

Table of primitive binary polynomials. II

For those n &gt; 5000 n &gt; 5000 , for which the factorization of 2 n − 1 {2^n} - 1 is known, the first primitive trinomial (if such exists) and a randomly generated primitive 5- and 7-nomial of degree n in G F ( 2 ) {\text {G}}F(2) are given, if the respective entry is absent from the previously published table.

Sojourn time test for maximum-length linearly recurring sequences with characteristic primitive trinomials

Sojourn time test for maximum-length linearly recurring sequences with characteristic primitive trinomials

Twisted GFSR generators

The generalized feed back shift register (GFSR) algorithm suggested by Lewis and Payne is a widely used pseudorandom number generator, but has the following serious drawbacks: (1) an initialization scheme to assure higher order equidistribution is involved and is time consuming; (2) each bit of the generated words constitutes an m -sequence based on a primitive trinomials, which shows poor randomness with respect to weight distribution; (3) a large working area is necessary; (4) the period of sequence is far shorter than the theoretical upper bound. This paper presents the twisted GFSR (TGFSR) algorithm, a slightly but essentially modified version of the GFSR, which solves all the above problems without loss of merit. Some practical TGFSR generators were implemented and passed strict empirical tests. These new generators are most suitable for simulation of a large distributive system, which requires a number of mutually independent pseudorandom number generators with compact size.

NEW PRIMITIVE TRINOMIALS OF MERSENNE-EXPONENT DEGREES FOR RANDOM-NUMBER GENERATION

The list of primitive binary trinomials with a degree equal to a Mersenne exponent is extended. The newly found primitive trinomials have a degree equal to the 29th and 30th Mersenne exponent. These trinomials enable the construction of new, high-performance random-number generators for use in large-scale Monte Carlo simulations.

Primitive 𝑡-nomials (𝑡=3,5) over 𝐺𝐹(2) whose degree is a Mersenne exponent ≤44497

All of the primitive trinomials over G F ( 2 ) GF(2) with degree p given by one of the Mersenne exponents 19937, 21701, 23209, and 44497 are presented. Also, one example of a primitive pentanomial over G F ( 2 ) GF(2) is presented for each degree up to 44497 that is a Mersenne exponent. The sieve used is briefly described. A problem is posed which conjectures the number of primitive pentanomials of degree p.

Random number generation with the recursion Xt = Xt−3 p ⊕ Xt−3 q

A generalized feedback shift register (GFSR) algorithm proposed by Lewis and Payne (1973) uses a primitive trinomial to generate a sequence of pseudorandom numbers. We propose a similar algorithm which uses a primitive polynomial with many nonzero terms, but generates a number as fast as the original GFSR algorithm. Our sequence is guaranteed to be equidistributed in higher dimensions and to have a good autocorrelation property. Extensive statistical tests have been performed on the sequences generated by our algorithm and the results were quite satisfactory.

Design considerations for Parallel pseudoRandom Pattern Generators

The design of a Parallel (pseudo)Random Pattern Generator (PRPG) based on a Linear Feedback Shift Register (LFSR) involves the selection of the number of stages (degree of the characteristic polynomial), selection of the feedback taps (the characteristic polynomial), and selection of a phaseshift network to remove the effects of correlations that exist between the bit streams from adjacent stages of the LFSR. The design considerations for decimated sequences are discussed, as is linear dependency density. A new phaseshift network is presented which uses only one 2-way EXCLUSIVE-OR per output. Appendixes contain a listing of the prime factors of the sequence length, an exhaustive list of primitive trinomials to degree 100, and a selected list of primitive polynomials that can be used to realize LFSRs with only two 2-way EXCLUSIVE-ORs.