Divisibility of Trinomials by Irreducible Polynomials over F_2
نویسندگان
چکیده
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 polynomials over F2 is established. And we extend Welch’s criterion for testing if an irreducible polynomial divides trinomials xm + xs + 1 to the trinomials xam + xbs + 1. Mathematics Subject Classification: 11T06; 12E05; 12E20; 13A05
منابع مشابه
Divisibility of polynomials over finite fields and combinatorial applications
Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f . Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polyn...
متن کاملA Necessary Condition of the Divisibility of Trinomials xam + xbs + 1 by any Irreducible Polynomial of Degree r over GF ( 2 )
In this paper we are interested in to the divisibility of trinomials of type x am + x bs + 1 over a finite field GF (2). We show that, for any irreducible polynomial T of degree r over GF (2) and for any non-zero integers a and b, if there exist integers m and s such that T divides x am + x bs + 1 then a and b are not divisible by (2 r − 1).
متن کاملAlgorithms for Finding Almost Irreducible and Almost Primitive Trinomials
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 ...
متن کاملAn Algorithm for Generating Irreducible Cubic Trinomials over Prime Field
This paper proposes an algorithm for generating irreducible cubic trinomials in the form x + ax + b, b ∈ Fp, where a is a certain fixed non-zero element in the prime field Fp. The proposed algorithm needs a certain irreducible cubic trinomial over Fp to be previously given as a generator; however, the proposed algorithm can generate irreducible cubic polynomials one after another by changing a ...
متن کاملPolynomial Residue Number System GF(2m) multiplier using trinomials
This paper introduces a new approach for implementing GF(2 m ) multiplication using Polynomial Residue Number Systems (PRNS). Irreducible trinomials are selected as the generating polynomials for the PRNS channels to enable conversion to-and-from PRNS to be implemented using simple XOR networks. A novel approach for modular reduction over GF(2 m ) is also presented for the PRNS architecture to ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2007