Faster implementation of ηt pairing over GF(3m) using minimum number of logical instructions for GF(3)-addition

Yuto Kawahara, Kazumaro Aoki, Tsuyoshi Takagi

Research output: Chapter in Book/Report/Conference proceedingConference contribution

17 Citations (Scopus)


The η T pairing in characteristic three is implemented by arithmetic in GF(3)={0,1,2}. Harrison et al. reported an efficient implementation of the GF(3)-addition by using seven logical instructions (consisting of AND, OR, and XOR) with the two-bit encoding { (0,0) 0, (0,1) 1, (1,0) 2 }. It has not yet been proven whether seven is the minimum number of logical instructions for the GF(3)-addition. In this paper, we show many implementations of the GF(3)-addition using only six logical instructions with different encodings such as { (1,1) 0, (0,1) 1, (1,0) 2 } or { (0,0) 0, (0,1) 1, (1,1) 2 }. We then prove that there is no implementation of the GF(3)-addition using five logical instructions with any encoding of GF(3) by two bits. Moreover, we apply the new GF(3)-additions to an efficient software implementation of the η T pairing. The running time of the η T pairing over GF(3509), that is considered to be realized as 128-bit security, using the new GF(3)-addition with the encoding { (0,0) 0, (0,1) 1, (1,1) 2 } is 16.3 milliseconds on an AMD Opteron 2.2-GHz processor. This is approximately 7% faster than the implementation using the previous GF(3)-addition with seven logical instructions.

Original languageEnglish
Title of host publicationPairing-Based Cryptography - Pairing 2008 - Second International Conference, Proceedings
Number of pages15
Volume5209 LNCS
Publication statusPublished - Sep 25 2008
Externally publishedYes
Event2nd International Conference on Pairing-Based Cryptography, Pairing 2008 - Egham, United Kingdom
Duration: Sep 1 2008Sep 3 2008

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume5209 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other2nd International Conference on Pairing-Based Cryptography, Pairing 2008
Country/TerritoryUnited Kingdom

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)


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