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Fast Generation of Elliptic Curve Base Points Using Efficient Exponentiation over $GF(p^m)$)  

Lee, Mun-Kyu (인하대학교 컴퓨터공학부)
Publication Information
Journal of KIISE:Computer Systems and Theory / v.34, no.3, 2007 , pp. 93-100 More about this Journal
Abstract
Since Koblitz and Miller suggested the use of elliptic curves in cryptography, there has been an extensive literature on elliptic curve cryptosystem (ECC). The use of ECC is based on the observation that the points on an elliptic curve form an additive group under point addition operation. To realize secure cryptosystems using these groups, it is very important to find an elliptic curve whose group order is divisible by a large prime, and also to find a base point whose order equals this prime. While there have been many dramatic improvements on finding an elliptic curve and computing its group order efficiently, there are not many results on finding an adequate base point for a given curve. In this paper, we propose an efficient method to find a random base point on an elliptic curve defined over $GF(p^m)$. We first show that the critical operation in finding a base point is exponentiation. Then we present efficient algorithms to accelerate exponentiation in $GF(p^m)$. Finally, we implement our algorithms and give experimental results on various practical elliptic curves, which show that the new algorithms make the process of searching for a base point 1.62-6.55 times faster, compared to the searching algorithm based on the binary exponentiation.
Keywords
Elliptic Curve; Parameter Generation; Base Point; Optimal Extension Field; Exponentiation;
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1 J.-C. Ha and S.-J. Moon, 'A common-multiplicand method to the Montgomery algorithm for speeding up exponentiation,' Information Processing Letters, Vol.66, pp.105-107, 1998   DOI   ScienceOn
2 E. F. Brickell, D. M. Gordon, K. S. McCurley, and D. B. Wilson, 'Fast exponentiation with precomputation,' Advances in Cryptology -Eurocrypt 92, LNCS, Vol.658, pp.200-207. Springer, 1993
3 C. H. Lim and P. J. Lee, 'More flexible exponentiation with precomputation,' Advances in Cryptology -CRYPTO 94, LNCS, Vol.839, pp.95-107. Springer, 1994
4 M. K. Lee, Y. Kim, K. Park, and Y. Cho, 'Efficient parallel exponentiation in $GF(q^n)$ using normal basis representations,' Journal of Algorithms, Vol.54, pp.205-221, 2005   DOI   ScienceOn
5 T. Kobayashi, '$Base-{\phi}$ method for elliptic curves of OEF,' IEICE Trans. Fundamentals, Vol.E83-A, No.4, pp.679-686, 2000
6 D. M. Gordon, 'A survey of fast exponentiation methods,' Journal of Algorithms, Vol.27, pp.129-146, 1998   DOI   ScienceOn
7 D. Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison-Wesley, Reading, Massachusetts, 3rd edition, 1998
8 J. Bos and M. Coster, 'Addition chain heuristics,' Advances in Cryptology- CRYPTO 89, LNCS, Vol.435, pp.400-407. Springer-Verlag, 1990
9 J.-C. Ha and S.-J. Moon, 'Fast exponentiation with common-multiplicand modular multiplication,' Journal of the Korea Information Science Society (C), Vol.3, No.5, pp.491-497, 1997
10 D. V. Bailey and C. Paar, 'Efficient arithmetic in finite field extensions with application in elliptic curve cryptography,' Journal of Cryptology, Vol.14, No.3, pp.153-176, 2001   DOI
11 N. P. Smart, 'A comparison of different finite fields for elliptic curve cryptosystems,' Computers and Mathematics with Applications, Vol.42, pp.91-100, 2001   DOI   ScienceOn
12 R. Schoof. 'Elliptic curves over finite fields and the computation of square roots mod p,' Mathematics of Computation, Vol.44, pp.483-494, 1985   DOI
13 G. B. Agnew, R. C. Mullin, and S. A. Vanstone, 'Fast exponentiation in $GF(2^n)$,' Advances in Cryptology-EUROCRYPT 88, LNCS, Vol.330, pp.251-256, Springer, 1988   DOI
14 J. von zur Gathen, 'Processor-efficient exponentiation in finite fields,' Information Processing Letters, Vol.41, pp.81-86, 1992   DOI   ScienceOn
15 TTAS.KO-12.0015, Digital Signature Mechanism with Appendix- Part 3: Korean Certificate-based Digital Signature Algorithm using Elliptic Curves, 2001
16 R. Lercier and F. Morain, 'Counting the number of points on elliptic curves over finite fields: strategies and performance,' Advances in Cryptology-Eurocrypt 95, LNCS, Vol.921, pp.79-94. Springer, 1995
17 R. Lercier, 'Finding good random elliptic curves for cryptosystems defined over $F_2$,' Advances in Cryptology-Eurocrypt 97, LNCS, Vol.1233, pp.379-392. Springer, 1997
18 D. V. Bailey and C. Paar, 'Optimal extension fields for fast arithmetic in public-key algorithms,' Advances in Cryptology- CRYPTO 98, LNCS, Vol.1462, pp.472-485. Springer, 1998   DOI   ScienceOn
19 N. Koblitz, 'Elliptic Curve Cryptosystems,' Mathematics of Computation, vol. 48, pp. 203-209, 1987   DOI
20 V. Miller. 'Use of elliptic curves in cryptography,' Advances in Cryptology- CRYPTO 85, LNCS, Vol. 218, pp.417-428, Springer-Verlag, 1986
21 IEEE P1363-2000, IEEE Standard Specifications for Public-Key Cryptography, 2000