Typical primitive polynomials over integer residue rings

Authors:
Tian Tian;Wen-Feng Qi
Affiliations:
Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, PR China;Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou, PR China and State Key Laboratory of Information Security (Institute of Software, Chinese Acad ...
Venue:
Finite Fields and Their Applications
Year:
2009

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Abstract

Let N be a product of distinct prime numbers and Z/(N) be the integer residue ring modulo N. In this paper, a primitive polynomial f(x) over Z/(N) such that f(x) divides x^s-c for some positive integer s and some primitive element c in Z/(N) is called a typical primitive polynomial. Recently typical primitive polynomials over Z/(N) were shown to be very useful, but the existence of typical primitive polynomials has not been fully studied. In this paper, for any integer m=1, a necessary and sufficient condition for the existence of typical primitive polynomials of degree m over Z/(N) is proved.