This Article 
 Bibliographic References 
 Add to: 
Hybrid Binary-Ternary Number System for Elliptic Curve Cryptosystems
February 2011 (vol. 60 no. 2)
pp. 254-265
Jithra Adikari, University of Calgary, Calgary
Vassil S. Dimitrov, University of Calgary, Calgary
Laurent Imbert, University of Calgary, Calgary and Université Montpellier, Montpellier
Single and double scalar multiplications are the most computational intensive operations in elliptic curve based cryptosystems. Improving the performance of these operations is generally achieved by means of integer recoding techniques, which aim at minimizing the scalars' density of nonzero digits. The hybrid binary-ternary number system provides both short representations and small density. In this paper, we present three novel algorithms for both single and double scalar multiplication. We present a detailed theoretical analysis, together with timings and fair comparisons over both tripling-oriented Doche-Ichart-Kohel curves and generic Weierstrass curves. Our experiments show that our algorithms are almost always faster than their widely used counterparts.

[1] N. Koblitz, "Elliptic Curve Cryptosystems," Math. of Computation, vol. 48, pp. 203-209, 1987.
[2] V.S. Miller, "Use of Elliptic Curves in Cryptography," Proc. Conf. Advances in Cryptology (CRYPTO' 85), pp. 417-426, 1986.
[3] I.F. Blake, G. Seroussi, and N.P. Smart, Elliptic Curves in Cryptography, first ed. Cambridge Univ. Press, 2000.
[4] L.C. Washington, Elliptic Curves: Number Theory and Cryptography, first ed. Chapman & Hall/CRC, May 2003.
[5] H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, and F. Vercauteren, Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC, July 2005.
[6] W. Diffie and M. Hellman, "New Directions in Cryptography," IEEE Trans. Information Theory, vol. IT-22, no. 6, pp. 644-654, Nov. 1976.
[7] R.L. Rivest, A. Shamir, and L. Adleman, "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems," Comm. ACM, vol. 21, pp. 120-126, 1978.
[8] A.D. Booth, "A Signed Binary Multiplication Technique," Quarterly J. Mechanics and Applied Math., vol. 4, no. 2, pp. 236-240, 1951.
[9] G.W. Reitwiesner, "Binary Arithmetic," Advances in Computers, vol. 1, pp. 231-308, Academic Press, 1960.
[10] D.E. Knuth, The Art of Computer Programming Vol. 2: Seminumerical Algorithms, Addsion Weslsey Longman Publishing Group, May 1969.
[11] M. Joye and S.-M. Yen, "Optimal Left-to-Right Binary Signed-Digit Exponent Recoding," IEEE Trans. Computers, vol. 49, no. 7, pp. 740-748, July 2000.
[12] D. Hankerson, A. Menezes, and S. Vanstone, Guide to Elliptic Curve Cryptography. Springer, 2004.
[13] V.S. Dimitrov, L. Imbert, and P.K. Mishra, "Efficient and Secure Elliptic Curve Point Multiplication Using Double-Base Chains," Proc. Int'l Conf. Theory and Applications of Cryptology and Information Security (ASIACRYPT '05), pp. 59-78, 2005.
[14] V.S. Dimitrov, L. Imbert, and P.K. Mishra, "The Double-Base Number System and Its Application to Elliptic Curve Cryptography," Math. of Computation, vol. 77, no. 262, pp. 1075-1104, 2008.
[15] C. Doche and L. Imbert, "Extended Double-Base Number System with Applications to Elliptic Curve Cryptography," Proc. Conf. Progress in Cryptology (INDOCRYPT '06), pp. 335-348, Dec. 2006.
[16] F. Morain and J. Olivos, "Speeding Up the Computations on an Elliptic Curve Using Addition-Subtraction Chains," Theoretical Informatics and Applications, vol. 24, pp. 531-543, 1990.
[17] H. Cohen, A. Miyaji, and T. Ono, "Efficient Elliptic Curve Exponentiation Using Mixed Coordinates," Proc. Int'l Conf. Theory and Applications of Cryptology and Information Security (ASIACRYPT '98), pp. 51-65, 1998.
[18] M. Ciet and F. Sica, "An Analysis of Double Base Number Systems and a Sublinear Scalar Multiplication Algorithm," Proc. Conf. Progress in Cryptology (MYCRYPT '05), pp. 171-182, 2005.
[19] R. Avanzi, V.S. Dimitrov, C. Doche, and F. Sica, "Extending Scalar Multiplication Using Double Bases," Proc. Int'l Conf. Theory and Applications of Cryptology and Information Security (ASIACRYPT '06), p. 130, 2006.
[20] N. Méloni and M.A. Hasan, "Elliptic Curve Point Scalar Multiplication Combining Yao's Algorithm and Double Bases," Proc. Workshop Cryptographic Hardware and Embedded Systems, Sept. 2009.
[21] P. Longa, "Accelerating the Scalar Multiplication on Elliptic Curve Cryptosystems over Prime Fields," Master's thesis, Univ. of Ottawa, 2007.
[22] P. Longa and C. Gebotys, "Fast Multibase Methods and Other Several Optimizations for Elliptic Curve Scalar Multiplication," Proc. Conf. Public Key Cryptography (PKC '09) pp. 443-462, 2009.
[23] J.A. Solinas, "Low-Weight Binary Representations for Pairs of Integers," Research Report CORR 2001-41, Center for Applied Cryptographic Research, Univ. of Waterloo, 2001.
[24] C. Doche, D.R. Kohel, and F. Sica, "Double-Base Number System for Multi-Scalar Multiplications," Proc. 28th Ann. Int'l Conf. Theory and Applications of Cryptographic Techniques (EUROCRYPT '09), Apr. 2009.
[25] E.G. Straus, "Addition Chains of Vectors (Problem 5125)," Am. Math. Monthly, vol. 70, pp. 806-808, 1964.
[26] V.S. Dimitrov and T.V. Cooklev, "Two Algorithms for Modular Exponentiation Based on Nonstandard Arithmetics," IEICE Trans. Fundamentals of Electronics, Comm. and Computer Science, vol. E78-A, no. 1,special issue on cryptography and information security, pp. 82-87, Jan. 1995.
[27] J. Adikari, V.S. Dimitrov, and L. Imbert, "Hybrid Binary-Ternary Joint form and Its Application in Elliptic Curve Cryptography" Proc. 19th IEEE Symp. Computer Arithmetic (ARITH), pp. 76-83, June 2009.
[28] A.H. Koblitz, N. Koblitz, and A. Menezes, "Elliptic Curve Cryptography: The Serpentine Course of a Paradigm Shift," to be published in J. Number Theory.
[29] C. Doche, T. Icart, and D.R. Kohel, "Efficient Scalar Multiplication by Isogeny Decompositions," Proc. Conf. Public Key Cryptography (PKC '06), pp. 191-206, 2006.
[30] D.J. Bernstein and T. Lange, "Analysis and Optimization of Elliptic-Curve Single-Scalar Multiplication," Finite Fields and Applications, Contemporary Mathematics, vol. 461, pp. 1-19, Am. Math. Soc., 2008.
[31] D.J. Bernstein and T. Lange, "Explicit-Formulas Database," Joint Work by D.J. Bernstein and T. Lange, Building on Work by Many Authors. http://www.hyperelliptic.orgEFD/, 2010.
[32] M. Ciet, M. Joye, K. Lauter, and P.L. Montgomery, "Trading Inversions for Multiplications in Elliptic Curve Cryptography," Designs, Codes and Cryptography, vol. 39, no. 2, pp. 189-206, May 2006.
[33] V.S. Dimitrov, G.A. Jullien, and W.C. Miller, "Theory and Applications of the Double-Base Number System," IEEE Trans. Computers, vol. 48, no. 10, pp. 1098-1106, Oct. 1999.
[34] V.S. Dimitrov and G.A. Jullien, "Loading the Bases: A New Number Representation with Applications," IEEE Circuits and Systems Magazine, vol. 3, no. 2, pp. 6-23, Nov. 2003.
[35] V.S. Dimitrov, G.A. Jullien, and W.C. Miller, "An Algorithm for Modular Exponentiation," Information Processing Letters, vol. 66, no. 3, pp. 155-159, 1998.
[36] V.S. Dimitrov, G.A. Jullien, and W.C. Miller, "Theory and Applications for a Double-Base Number System," Proc. 13th Symp. Computer Arithmetic (ARITH), p. 44, July 1997.
[37] C. Doche and L. Habsieger, "A Tree-Based Approach for Computing Double-Base Chains," Proc. 13th Australasian Conf. Information Security and Privacy (ACISP '08), pp. 433-446, 2008.
[38] D.J. Bernstein, P. Birkner, T. Lange, and C. Peters, "Optimizing Double-Base Elliptic-Curve Single-Scalar Multiplication," Proc. Conf. Progress in Cryptology (INDOCRYPT '07), pp. 167-182, 2007.
[39] Nat'l Inst. of Standards and Tech nology, FIPS 186-2, Digital Signature Standard, Fed. Information Processing Standards Publication, fips186-2-change1.pdf, 2000.
[40] C. Doche, "Tripling Oriented DIK Curve Software Implementations," Personal Communication, Sept. 2008.
[41] K. Okeya and K. Sakurai, "Use of Montgomery Trick in Precomputation of Multi-Scalar Multiplication in Elliptic Curve Cryptosystems," Trans. Fundamentals of Electronics, Comm. and Computer Science, vol. E86-A, no. 1, pp. 98-112, 2003.
[42] H. Cohen, A Course in Computational Algebraic Number Theory, third ed. Springer, Sept. 1993.
[43] Free Software Foundation, "GMP, Arithmetic without Limitations," http:/, Apr. 2009.

Index Terms:
Elliptic curve cryptography, single/double scalar multiplication, hybrid binary-ternary number system, DIK-3 curves.
Jithra Adikari, Vassil S. Dimitrov, Laurent Imbert, "Hybrid Binary-Ternary Number System for Elliptic Curve Cryptosystems," IEEE Transactions on Computers, vol. 60, no. 2, pp. 254-265, Feb. 2011, doi:10.1109/TC.2010.138
Usage of this product signifies your acceptance of the Terms of Use.