Reduction optimal trinomials for efficient software implementation of the ηT pairing

Toshiya Nakajima, Tetsuya Izu, Tsuyoshi Takagi

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

2 Citations (Scopus)


The ηT pairing for supersingular elliptic curve over GF(3m) has been paid attention because of its computational efficiency. Since most parts of computation of the ηT pairing are multiplications over GF(3m), it is important to improve the speed of the multiplication when implementing the ηT pairing. In this paper we consider software implementation of multiplication over GF (3 m) and propose to use irreducible trinomials xm + ax k + b over GF(3) such that w, bit length of word of targeted CPU, divides k. We call the trinomials "reduction optimal trinomials (ROTs)". ROTs actually exist for several m's and typical values of w = 16 and 32. We list them for extension degrees m = 97, 167, 193 and 239. These m's are derived from security considerations. Using ROT it is possible to implement efficient modulo operation (reduction) in multiplication over GF(3m) comparing with the case using other type of trinomials (e.g., trinomials with minimum k for each m). The reason of this is that for the cases of reduction by ROT the number of shift operations on multiple precision data reduces to less than half comparing with the cases by other trinomials. Implementation results show that reduction algorithm specialized for ROT is 20-30% faster on 32-bit CPU and around 40% faster on 16-bit CPU than algorithm for irreducible trinomials with general k.

Original languageEnglish
Title of host publicationAdvances in Information and Computer Security - Second International Workshop on Security, IWSEC 2007, Proceedings
PublisherSpringer Verlag
Number of pages14
ISBN (Print)9783540756507
Publication statusPublished - 2007
Externally publishedYes
Event2nd International Workshop on Security, IWSEC 2007 - Nara, Japan
Duration: Oct 29 2007Oct 31 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4752 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other2nd International Workshop on Security, IWSEC 2007

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science


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