منابع مشابه
On Asymptotic Behavior of the Ratio Between the Numbers of Binary Primitive and Irreducible Polynomials
In this paper we study the ratio θ(n) = λ2(n) ψ2(n) , where λ2(n) is the number of primitive polynomials and ψ2(n) is the number of irreducible polynomials in GF (2)[x] of degree n. Let n = Q` i=1 p ri i be the prime factorization of n, where pi are odd primes. We show that θ(n) tends to 1 and θ(2n) is asymptotically not less than 2/3 when ri are fixed and pi tend to infinity. We also, describe...
متن کاملOn the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials
In this paper, we study the ratio θ(n) = λ2(n) ψ2(n) , where λ2(n) is the number of primitive polynomials and ψ2(n) is the number of irreducible polynomials in GF (2)[x] of degree n. Let n = ∏̀ i=1 p i be the prime factorization of n. We show that, for fixed l and ri, θ(n) is close to 1 and θ(2n) is not less than 2/3 for sufficiently large primes pi. We also describe an infinite series of values...
متن کاملOn Fast Multiplication in Binary Finite Fields and Optimal Primitive Polynomials over GF(2)
In this paper we present a number of algorithms and optimizationtechniques to speedup computations in binary extension fields overGF(2). Particularly, we consider multiplication and modular reduc-tion solutions. Additionally, we provide the table of optimal binaryprimitive polynomials over GF(2) of degree 2 ≤ d < 2048, and theclass of functions for optimal modular reduction ...
متن کاملTwelve New Primitive Binary Trinomials
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...
متن کاملTen 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. 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...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1973
ISSN: 0025-5718
DOI: 10.2307/2005535