Generating prime degree irreducible polynomials by using irreducible all‐one polynomial over F2

Abstract

AbstractIn most of the methods of public key cryptography devised in recent years, a finite field of a large order is used as the field of definition. In contrast, there are many studies in which a higher‐degree extension field of characteristic 2 is fast implemented for easier hardware realization. There are also many reports of the generation of the required higher‐degree irreducible polynomial, and of the construction of a basis suited to fast implementation, such as an optimal normal basis (ONB). For generating higher‐degree irreducible polynomials, there is a method in which a 2m‐th degree self‐reciprocal irreducible polynomial is generated from an m‐th degree irreducible polynomial by a simple polynomial transformation (called the self‐reciprocal transformation). This paper considers this transformation and shows that when the set of zeros of the m‐th degree irreducible polynomial forms a normal basis, the set of zeros of the generated 2m‐th order self‐reciprocal irreducible polynomial also forms a normal base. Then it is clearly shown that there is a one‐to‐one correspondence between the transformed irreducible polynomial and the generated self‐reciprocal irreducible polynomial. Consequently, the inverse transformation of the self‐reciprocal transformation (self‐reciprocal inverse transformation) can be applied to a self‐reciprocal irreducible polynomial. It is shown that an m‐th degree irreducible polynomial can always be generated from a 2m‐th degree self‐reciprocal irreducible polynomial by the self‐reciprocal inverse transformation. We can use this fact for generating 1/2‐degree irreducible polynomials. As an application of 1/2‐degree irreducible polynomial generation, this paper proposes a method which generates a prime degree irreducible polynomial with a Type II ONB as its zeros. © 2005 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 88(7): 23–32, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20151