### Abstract

A square root (SQRT) algorithm in GF(p^{m}) (m = r _{0}r_{1}⋯ r_{n-1}-12^{d}, r_{i}: odd prime, d > 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in GF(p ^{2d} ), where most of the computations are performed in the corresponding subfields GF(p^{2i} ) for 0 ≤ i ≤ d-1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF(p ^{m}) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the Tonelli-Shanks algorithm and the proposed algorithm in GF(p^{22}), GF(p^{44}) and GF(p ^{88}) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF(p ^{22}), by 45 times in GF(p^{44}), and by 70 times in GF(p ^{88}), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 94-106 |

Number of pages | 13 |

Volume | 4296 LNCS |

Publication status | Published - 2006 |

Event | ICISC 2006: 9th International Conference on Information Security and Cryptology - Busan, Korea, Republic of Duration: Nov 30 2006 → Dec 1 2006 |

### Other

Other | ICISC 2006: 9th International Conference on Information Security and Cryptology |
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Country | Korea, Republic of |

City | Busan |

Period | 11/30/06 → 12/1/06 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 4296 LNCS, pp. 94-106)

**A high-speed square root algorithm in extension fields.** / Katou, Hidehiro; Wang, Feng; Nogami, Yasuyuki; Morikawa, Yoshitaka.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 4296 LNCS, pp. 94-106, ICISC 2006: 9th International Conference on Information Security and Cryptology, Busan, Korea, Republic of, 11/30/06.

}

TY - GEN

T1 - A high-speed square root algorithm in extension fields

AU - Katou, Hidehiro

AU - Wang, Feng

AU - Nogami, Yasuyuki

AU - Morikawa, Yoshitaka

PY - 2006

Y1 - 2006

N2 - A square root (SQRT) algorithm in GF(pm) (m = r 0r1⋯ rn-1-12d, ri: odd prime, d > 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in GF(p 2d ), where most of the computations are performed in the corresponding subfields GF(p2i ) for 0 ≤ i ≤ d-1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF(p m) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the Tonelli-Shanks algorithm and the proposed algorithm in GF(p22), GF(p44) and GF(p 88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF(p 22), by 45 times in GF(p44), and by 70 times in GF(p 88), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.

AB - A square root (SQRT) algorithm in GF(pm) (m = r 0r1⋯ rn-1-12d, ri: odd prime, d > 0: integer) is proposed in this paper. First, the Tonelli-Shanks algorithm is modified to compute the inverse SQRT in GF(p 2d ), where most of the computations are performed in the corresponding subfields GF(p2i ) for 0 ≤ i ≤ d-1. Then the Frobenius mappings with an addition chain are adopted for the proposed SQRT algorithm, in which a lot of computations in a given extension field GF(p m) are also reduce to those in a proper subfield by the norm computations. Those reductions of the field degree increase efficiency in the SQRT implementation. More specifically the Tonelli-Shanks algorithm and the proposed algorithm in GF(p22), GF(p44) and GF(p 88) were implemented on a Pentium4 (2.6 GHz) computer using the C++ programming language. The computer simulations showed that, on average, the proposed algorithm accelerates the SQRT computation by 25 times in GF(p 22), by 45 times in GF(p44), and by 70 times in GF(p 88), compared to the Tonelli-Shanks algorithm, which is supported by the evaluation of the number of computations.

UR - http://www.scopus.com/inward/record.url?scp=34547443654&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547443654&partnerID=8YFLogxK

M3 - Conference contribution

SN - 3540491120

SN - 9783540491125

VL - 4296 LNCS

SP - 94

EP - 106

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -