# A General Methodology for Pipelining the Point Multiplication Operation in Curve Based Cryptography

• Kishan Chand Gupta
• Pinakpani Pal
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3989)

## Abstract

Pipelining is a well-known performance enhancing technique in computer science. Point multiplication is the computationally dominant operation in curve based cryptography. It is generally computed by repeatedly invoking some curve (group) operation like doubling, tripling, halving, addition of group elements. Such a computational procedure may be efficiently computed in a pipeline. More generally, let Π be a computational procedure, which computes its output by repeatedly invoking processes from a set of similar processes. Employing pipelining technique may speed up the running time of the computational procedure. To find pipeline sequence by trial and error method is a nontrivial task. In the current work, we present a general methodology, which given any such computational procedure Π can find a pipelined version with improved computational speed. To our knowledge, this is the first such attempt in curve based cryptography, where it can be used to speed up the point multiplication methods using inversion-free explicit formula for curves over prime fields. As an example, we employ the proposed general methodology to derive a pipelined version of the hyperelliptic curve binary algorithm for point multiplication and obtain a performance gain of 32% against the ideal theoretical value of 50%.

## Keywords

Point Multiplication Elliptic Curve Hyperelliptic Curve Elliptic Curve Cryptography Discrete Logarithm Problem

## References

1. 1.
Avanzi, R.M.: Countermeasures against Differential Power Analysis for Hyperelliptic Curve Cryptosystems. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 366–381. Springer, Heidelberg (2003)
2. 2.
Avanzi, R., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: In: Frey, G., Cohen, H. (eds.) Handbook of Elliptic and Hyperelliptic Curve Cryptography. CRC Press, Boca Raton (2005)Google Scholar
3. 3.
Bertoni, G., Breveglieri, L., Wollinger, T., Paar, C.: Finding Optimum Parallel Coprocessor Design for Genus 2 Hyperelliptic Curve Cryptosystems. Cryptology ePrint Archive, Report 2004/29 (2004), http://eprint.iacr.org/
4. 4.
Cantor, D.G.: Computing in the Jacobian of a Hyperelliptic curve. Mathematics of Computation 48, 95–101 (1987)
5. 5.
Chevallier-Mames, B., Ciet, M., Joye, M.: Low-cost Solutions for Preventing Simple Side-Channel Analysis: Side-Channel Atomicity. IEEE Trans. on Computers 53, 760–768 (2004)
6. 6.
Cohen, H., Miyaji, A., Ono, T.: Efficient Elliptic Curve Exponentiation Using Mixed Coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)
7. 7.
Flon, S., Oyono, R.: Fast Arithmetic on Jacobians of Picard Curves. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 55–68. Springer, Heidelberg (2004)
8. 8.
Galbraith, S.D., Paulus, S.M., Smart, N.P.: Arithmatic on Superelliptic Curves. Mathematics of Computations 71(237), 393–405 (2002)
9. 9.
Gaudry, P., Harley, R.: Counting Points on Hyperelliptic Curves over Finite Fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 297–312. Springer, Heidelberg (2000)
10. 10.
Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography, Springer-Verlag Professional Computing Series (2004) ISBN: 0-387-95273-XGoogle Scholar
11. 11.
Harley, R.: Fast Arithmetic on Genus 2 Curves, Available at: http://cristal.inria.fr/~harley/hyper/adding.txt
12. 12.
Joye, M., Tymen, C.: Protections against Differential Analysis for Elliptic Curve Cryptography. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 402–410. Springer, Heidelberg (2001)
13. 13.
Koblitz, N.: Elliptic Curve Cryptosystems. Mathematics of Computations 48, 203–209 (1987)
14. 14.
Koblitz, N.: Hyperelliptic Cryptosystems. Journal of Cryptology 1(3), 139–150 (1989)
15. 15.
Koblitz, N.: CM-Curves with Good Cryptographic Properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992)Google Scholar
16. 16.
Kocher, P.C.: Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS, and Other Systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)Google Scholar
17. 17.
Kocher, P.C., Jaffe, J., Jun, B.: Differential Power Analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)
18. 18.
Lange, T.: Efficient Arithmetic on Genus 2 Curves over Finite Fields via Explicit Formulae. Cryptology ePrint Archive, Report 2002/121 (2002), http://eprint.iacr.org/
19. 19.
Lange, T.: Inversion-free Arithmetic on Genus 2 Hyperelliptic Curves. Cryptology ePrint Archive, Report 2002/147 (2002), http://eprint.iacr.org/
20. 20.
Lange, T.: Weighted Co-ordinates on Genus 2 Hyperelliptic Curves. Cryptology ePrint Archive, Report 2002/153 (2002), http://eprint.iacr.org/
21. 21.
Lange, T.: Formulae for Arithmetic on Genus 2 Hyperelliptic Curves. J. AAECC (to appear, 2004), http://www.itsc.ruhr-uni-bochum.de/tanja/preprints.html
22. 22.
Menezes, A.J., Vanstone, S.: Elliptic curve cryptosystems and their implementation. Journal of Cryptology 6, 209–224 (1993)
23. 23.
Menezes, A., Wu, Y., Zuccherato, R.: An Elementary Introduction to Hyperelliptic Curves. Technical Report, CORR 96-19, University of Waterloo, Canada (1996), Available at: http://www.cacr.math.uwaterloo.ca
24. 24.
Miller, V.S.: Use of Elliptic Curves in Cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
25. 25.
Mishra, P.K.: Pipelined Computation of Scalar Multiplication in Elliptic Curve Cryptosystems. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 328–342. Springer, Heidelberg (2004) (Full version to appear in IEEE Trans. on Computers)
26. 26.
Mishra, P.K., Sarkar, P.: Parallelizing Explicit Formula for Arithmetic in the Jacobian of Hyperelliptic Curves. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 93–110. Springer, Heidelberg (2003)
27. 27.
Schroeppel, R.: Elliptic curve point halving wins big. In: Second Midwest Arithmetical Geometry in Cryptography Workshop, Urbana, Illinois (November 2000)Google Scholar
28. 28.
Solinas, J.A.: An Improved Algorithm for Arithmetic on a Family of Elliptic Curves. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 357–371. Springer, Heidelberg (1997)
29. 29.
Spallek, A.M.: Kurven vom Geschletch 2 und irhe Anwendung in Public-Key-Kryptosystemen. PhD Thesis, Universität Gesamthochschule, Essen (1994)Google Scholar

## Authors and Affiliations

• Kishan Chand Gupta
• 1