Systematic Generation of An Irreducible Polynomial of An Arbitrary Degree m over F p Such That p > m

There are many studies for generating irreducible polynomials (L. M. Adleman & H. W. Lenstra (1986)) − (Ian. F. Blake et al., (1993)). This is because irreducible polynomials play critical roles in the cases such as constructing extension field or generating random sequence. The problem of generating irreducible polynomial is theoretically interesting and have attracted many scientists and engineers. Those previous works are roughly classified by the objective: one is arbitrary − degree and the other is efficient for fast arithmetic operations in extension field. This paper is related to the former. As an application of the proposed method, the authors consider variable key − length public key cryptography (M. Scott (2006)). Adleman et al. (L. M. Adleman & H. W. Lenstra (1986)) have shown that an irreducible polynomial of degree m over Fp with an arbitrary pair of p and m is generated by using a Gauss period normal basis (GNB) in and Shoup shown almost the same idea (V. Shoup (1990)). Because, as introduced in Gao's paper (S. Gao (1993)), a GNB in always exists for an arbitrary pair of p and m such that 4p does not divide m(p − 1). However, they do not explicitly give a concrete generating algorithm. Of course, their calculation costs are not explicitly evaluated. Their methods are based on the minimal polynomial determination and efficiently using Newton's formula (R. Lidl & H. Niederreiter (1984)). On the other hand, the authors (K. Makita et al., (2005)) have explicitly given efficient generating algorithms in which characteristic p = 2 is only dealt with. These algorithms (K. Makita et al., (2005)) determine the minimal polynomial of TypeII ONB in quite fast; however, if TypeII ONB does not exist in , it does not work. Thus, our previous works restrict not only degrees but also the characteristic to 2. Using Newton's formula and a certain special class of Gauss period normal bases in , this paper gives a concrete algorithm that efficiently generates an irreducible polynomial of degree m over Fp for an arbitrary pair of m and p > m. When p > m, it is automatically satisfied that 4p does not divide m(p − 1). The restriction p > m