### 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 |
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Pages (from-to) | 23-32 |

Number of pages | 10 |

Journal | Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi) |

Volume | 88 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jul 1 2005 |

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### Keywords

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

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

_{2}.

*Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)*,

*88*(7), 23-32. https://doi.org/10.1002/ecjc.20151