### Abstract

The η_{T} pairing for supersingular elliptic curves over GF(3^{m}) has been paid attention because of its computational efficiency. Since most computation parts of the η_{T} pairing are GF(3^{m}) multiplications, it is important to improve the speed of the multiplication when implementing the η_{T} pairing. In this paper we investigate software implementation of GF(3^{m}) multiplication and propose using irreducible trinomials x^{m} +ax^{k}+b over GF(3) such that k is a multiple of w, where w is the bit length of the word of targeted CPU. We call the trinomials "reduction optimal trinomials (ROTs)." ROTs actually exist for several m's and for typical values of w - 16 and 32. We list them for extension degrees m = 97, 167, 193, 239, 317, and 487. These m's are derived from security considerations. Using ROTs, we are able to implement efficient modulo operations (reductions) for GF(3^{m}) multiplication compared with cases in which other types of irreducible trinomials are used (e.g., trinomials with a minimum k for each m). The reason for this is that for cases using ROTs, the number of shift operations on multiple precision data is reduced to less than half compared with cases using other trinomials. Our implementation results show that programs of reduction specialized for PLOTs are 20-30% faster on 32-bit CPU and approximately 40% faster on 16-bit CPU compared with programs using irreducible trinomials with general k.

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

Pages (from-to) | 2379-2386 |

Number of pages | 8 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E91-A |

Issue number | 9 |

DOIs | |

Publication status | Published - 2008 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Signal Processing
- Computer Graphics and Computer-Aided Design
- Electrical and Electronic Engineering
- Applied Mathematics

### Cite this

_{T}pairing.

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E91-A*(9), 2379-2386. https://doi.org/10.1093/ietfec/e91-a.9.2379

**Reduction optimal trinomials for efficient software implementation of the η _{T} pairing.** / Nakajima, Toshiya; Izu, Tetsuya; Takagi, Tsuyoshi.

Research output: Contribution to journal › Article

_{T}pairing',

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E91-A, no. 9, pp. 2379-2386. https://doi.org/10.1093/ietfec/e91-a.9.2379

_{T}pairing. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences. 2008;E91-A(9):2379-2386. https://doi.org/10.1093/ietfec/e91-a.9.2379

}

TY - JOUR

T1 - Reduction optimal trinomials for efficient software implementation of the ηT pairing

AU - Nakajima, Toshiya

AU - Izu, Tetsuya

AU - Takagi, Tsuyoshi

PY - 2008

Y1 - 2008

N2 - The ηT pairing for supersingular elliptic curves over GF(3m) has been paid attention because of its computational efficiency. Since most computation parts of the ηT pairing are GF(3m) multiplications, it is important to improve the speed of the multiplication when implementing the ηT pairing. In this paper we investigate software implementation of GF(3m) multiplication and propose using irreducible trinomials xm +axk+b over GF(3) such that k is a multiple of w, where w is the bit length of the word of targeted CPU. We call the trinomials "reduction optimal trinomials (ROTs)." ROTs actually exist for several m's and for typical values of w - 16 and 32. We list them for extension degrees m = 97, 167, 193, 239, 317, and 487. These m's are derived from security considerations. Using ROTs, we are able to implement efficient modulo operations (reductions) for GF(3m) multiplication compared with cases in which other types of irreducible trinomials are used (e.g., trinomials with a minimum k for each m). The reason for this is that for cases using ROTs, the number of shift operations on multiple precision data is reduced to less than half compared with cases using other trinomials. Our implementation results show that programs of reduction specialized for PLOTs are 20-30% faster on 32-bit CPU and approximately 40% faster on 16-bit CPU compared with programs using irreducible trinomials with general k.

AB - The ηT pairing for supersingular elliptic curves over GF(3m) has been paid attention because of its computational efficiency. Since most computation parts of the ηT pairing are GF(3m) multiplications, it is important to improve the speed of the multiplication when implementing the ηT pairing. In this paper we investigate software implementation of GF(3m) multiplication and propose using irreducible trinomials xm +axk+b over GF(3) such that k is a multiple of w, where w is the bit length of the word of targeted CPU. We call the trinomials "reduction optimal trinomials (ROTs)." ROTs actually exist for several m's and for typical values of w - 16 and 32. We list them for extension degrees m = 97, 167, 193, 239, 317, and 487. These m's are derived from security considerations. Using ROTs, we are able to implement efficient modulo operations (reductions) for GF(3m) multiplication compared with cases in which other types of irreducible trinomials are used (e.g., trinomials with a minimum k for each m). The reason for this is that for cases using ROTs, the number of shift operations on multiple precision data is reduced to less than half compared with cases using other trinomials. Our implementation results show that programs of reduction specialized for PLOTs are 20-30% faster on 32-bit CPU and approximately 40% faster on 16-bit CPU compared with programs using irreducible trinomials with general k.

UR - http://www.scopus.com/inward/record.url?scp=77953446458&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953446458&partnerID=8YFLogxK

U2 - 10.1093/ietfec/e91-a.9.2379

DO - 10.1093/ietfec/e91-a.9.2379

M3 - Article

AN - SCOPUS:77953446458

VL - E91-A

SP - 2379

EP - 2386

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 9

ER -