### Abstract

A pseudo-random sequence generated by using a primitive polynomial, trace function, and Legendre symbol has been researched in our previous work. Our previous sequence has some interesting features such as period, autocorrelation, and linear complexity. A pseudo-random sequence widely used in cryptography. However, from the aspect of the practical use in cryptographic systems sequence needs to generate swiftly. Our previous sequence generated by utilizing a primitive polynomial, however, finding a primitive polynomial requires high calculating cost when the degree or the characteristic is large. It's a shortcoming of our previous work. The main contribution of this work is to find some relation between the generated sequence and irreducible polynomials. The purpose of this relationship is to generate the same sequence without utilizing a primitive polynomial. From the experimental observation, it is found that there are (p - 1)/2 kinds of polynomial, which generates the same sequence. In addition, some of these polynomials are non-primitive polynomial. In this paper, these relationships between the sequence and the polynomials are shown by some examples. Furthermore, these relationships are proven theoretically also.

Original language | English |
---|---|

Pages (from-to) | 166-172 |

Number of pages | 7 |

Journal | Journal of Information and Communication Convergence Engineering |

Volume | 16 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2018 |

### Fingerprint

### Keywords

- Irreducible polynomial
- Legendre symbol
- Odd characteristic field
- Pseudo-random sequence
- Trace function

### ASJC Scopus subject areas

- Information Systems
- Media Technology
- Computer Networks and Communications
- Electrical and Electronic Engineering

### Cite this

*Journal of Information and Communication Convergence Engineering*,

*16*(3), 166-172. https://doi.org/10.6109/jicce.2018.16.3.166

**Relation between the irreducible polynomials that generates the same binary sequence over odd characteristic field.** / Ali, Md Arshad; Kodera, Yuta; Park, Taehwan; Kusaka, Takuya; Nogmi, Yasuyuki; Kim, Howon.

Research output: Contribution to journal › Article

*Journal of Information and Communication Convergence Engineering*, vol. 16, no. 3, pp. 166-172. https://doi.org/10.6109/jicce.2018.16.3.166

}

TY - JOUR

T1 - Relation between the irreducible polynomials that generates the same binary sequence over odd characteristic field

AU - Ali, Md Arshad

AU - Kodera, Yuta

AU - Park, Taehwan

AU - Kusaka, Takuya

AU - Nogmi, Yasuyuki

AU - Kim, Howon

PY - 2018/1/1

Y1 - 2018/1/1

N2 - A pseudo-random sequence generated by using a primitive polynomial, trace function, and Legendre symbol has been researched in our previous work. Our previous sequence has some interesting features such as period, autocorrelation, and linear complexity. A pseudo-random sequence widely used in cryptography. However, from the aspect of the practical use in cryptographic systems sequence needs to generate swiftly. Our previous sequence generated by utilizing a primitive polynomial, however, finding a primitive polynomial requires high calculating cost when the degree or the characteristic is large. It's a shortcoming of our previous work. The main contribution of this work is to find some relation between the generated sequence and irreducible polynomials. The purpose of this relationship is to generate the same sequence without utilizing a primitive polynomial. From the experimental observation, it is found that there are (p - 1)/2 kinds of polynomial, which generates the same sequence. In addition, some of these polynomials are non-primitive polynomial. In this paper, these relationships between the sequence and the polynomials are shown by some examples. Furthermore, these relationships are proven theoretically also.

AB - A pseudo-random sequence generated by using a primitive polynomial, trace function, and Legendre symbol has been researched in our previous work. Our previous sequence has some interesting features such as period, autocorrelation, and linear complexity. A pseudo-random sequence widely used in cryptography. However, from the aspect of the practical use in cryptographic systems sequence needs to generate swiftly. Our previous sequence generated by utilizing a primitive polynomial, however, finding a primitive polynomial requires high calculating cost when the degree or the characteristic is large. It's a shortcoming of our previous work. The main contribution of this work is to find some relation between the generated sequence and irreducible polynomials. The purpose of this relationship is to generate the same sequence without utilizing a primitive polynomial. From the experimental observation, it is found that there are (p - 1)/2 kinds of polynomial, which generates the same sequence. In addition, some of these polynomials are non-primitive polynomial. In this paper, these relationships between the sequence and the polynomials are shown by some examples. Furthermore, these relationships are proven theoretically also.

KW - Irreducible polynomial

KW - Legendre symbol

KW - Odd characteristic field

KW - Pseudo-random sequence

KW - Trace function

UR - http://www.scopus.com/inward/record.url?scp=85055900483&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055900483&partnerID=8YFLogxK

U2 - 10.6109/jicce.2018.16.3.166

DO - 10.6109/jicce.2018.16.3.166

M3 - Article

AN - SCOPUS:85055900483

VL - 16

SP - 166

EP - 172

JO - Journal of Information and Communication Convergence Engineering

JF - Journal of Information and Communication Convergence Engineering

SN - 2234-8255

IS - 3

ER -