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

Kei Makita, Yasuyuki Nogami, Tatsuo Sugimura

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

## Abstract

In 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.

Original language English 23-32 10 Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi) 88 7 https://doi.org/10.1002/ecjc.20151 Published - Jul 1 2005

## Keywords

• Elliptic curve cryptography
• Self-reciprocal irreducible polynomial
• Type I/Type II optimal normal basis

## ASJC Scopus subject areas

• Electrical and Electronic Engineering

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