Abstract
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.
| Original language | English |
|---|---|
| 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 |
|---|---|
| Country | Korea, Republic of |
| City | Busan |
| Period | 11/30/06 → 12/1/06 |
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ASJC Scopus subject areas
- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science
Cite this
A high-speed square root algorithm in extension fields. / Katou, Hidehiro; Wang, Feng; Nogami, Yasuyuki; Morikawa, Yoshitaka.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 4296 LNCS 2006. p. 94-106.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
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.
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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 -