Efficient exponentiation in extensions of finite fields without fast frobenius mappings

Yasuyuki Nogami, Hidehiro Kato, Kenta Nekado, Yoshitaka Morikawa

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


This paper proposes an exponentiation method with Frobenius mappings. The main target is an exponentiation in an extension field. This idea can be applied for scalar multiplication of a rational point of an elliptic curve defined over an extension field. The proposed method is closely related to so-called interleaving exponentiation. Unlike interleaving exponentiation methods, it can carry out several exponentiations of the same base at once. This happens in some pairing-based applications. The efficiency of using Frobenius mappings for exponentiation in an extension field was well demonstrated by Avanzi and Mihailescu. Their exponentiation method efficiently decreases the number of multiplications by inversely using many Frobenius mappings. Compared to their method, although the number of multiplications needed for the proposed method increases about 20%, the number of Frobenius mappings becomes small. The proposed method is efficient for cases in which Frobenius mapping cannot be carried out quickly.

Original languageEnglish
Pages (from-to)818-825
Number of pages8
JournalETRI Journal
Issue number6
Publication statusPublished - Dec 2008


  • Exponentiation
  • Extension field
  • Frobenius mapping
  • Modular polynomial
  • Prime field
  • Window method

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Computer Science(all)
  • Electrical and Electronic Engineering


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