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

Md Arshad Ali, Yuta Kodera, Taehwan Park, Takuya Kusaka, Yasuyuki Nogmi, Howon Kim

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)166-172
Number of pages7
JournalJournal of Information and Communication Convergence Engineering
Issue number3
Publication statusPublished - 2018


  • 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


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