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http://dx.doi.org/10.7840/kics.2012.37A.12.1031

Square and Cube Root Algorithms in Finite Field and Their Applications  

Cho, Gook Hwa (성균관대학교 수학과)
Ha, Eunhye (성균관대학교 수학과)
Koo, Namhun (성균관대학교 수학과)
Kwon, Soonhak (성균관대학교 수학과)
Publication Information
The Journal of Korean Institute of Communications and Information Sciences / v.37A, no.12, 2012 , pp. 1031-1037 More about this Journal
Abstract
We study an algorithm that can efficiently find square roots and cube roots by modifying Tonelli-Shanks algorithm, which has an application in Number Field Sieve (NFS). The Number Field Sieve, the fastest known factoring algorithm, is a powerful tool for factoring very large integer. NFS first chooses two polynomials having common root modulo N, and it consists of the following four major steps; 1. Polynomial Selection 2. Sieving 3. Matrix 4. Square Root. The last step of NFS needs the process of square root computation in Number Field, which can be computed via square root algorithm over finite field.
Keywords
NFS; Tonelli-Shanks algorithm; CRT; Finite Field;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
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1 E. Bach, "A note on square roots in finite fields," IEEE Trans. Inform. Theory vol. 36, no. 6, pp. 1494-1498, Oct. 1990.   DOI   ScienceOn
2 J. P. Buhler, H. W. Lenstra, and C. Pomerance, "Factoring integers with the number field sieve," Reprinted in The Development of the Number Field Sieve, Lecture Notes in Mathematics 1554. A.K. Lenstra, HW. Lenstra, Jr., Eds., Jun. 1993
3 J. Dreibelbis, Implementing the General Number Field Sieve, Rochester Institute of Technology, Jun. 2003.
4 D. G. Han, D. Choi, and H. Kim, "Improved computation of square roots in specific finite fields," IEEE Trans. Comput., vol. 58, no. 02, pp. 188-196, Feb. 2009.   DOI   ScienceOn
5 T. Kleinjung, K. Aoki, J. Franke, A. Lenstra, E. Thome, J. Bos, P. Gaudry, A. Kruppa, P. Montgomery, D. Osvik, H. te Riele, A. Timofeev, and P. Zimmermann, "Factorization of a 768-bit RSA modulus," in Proc. IACR Crypto, pp. 333-350, Aug. 2010.
6 F. Kong, Z. Cai, J. Yu, and D. Li, "Improved generalized Atkin algorithm for computing square roots in finite fields," Inform. Process. Lett., vol. 98, no. 1, pp. 1-5, April. 2006   DOI   ScienceOn
7 N. Nishihara, R. Harasawa, Y. Sueyoshi, and A. Kudo, "A remark on the computation of cube roots in finite fields," IACR Cryptology ePrint Archive, Sep. 2009
8 G. H. Jo, N. Koo, S. Kwon, "Two cubic polynomial selection for the number field sieve," J. KICS, vol. 36, no. 10. pp. 614-620, Oct. 2011   DOI