Abstract
It is known that every positive integer n can be represented as a finite sum of the form n = sum(a i 2i), where a i in {0,1,-1} for all i, and no two consecutive a i ’s are non-zero. Such sums are called nonadjacent representations. Nonadjacent representations are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications.
In this paper, we investigate if other digit sets of the form {0,1,x}, where x is an integer, provide each positive integer with a nonadjacent representation. If a digit set has this property we call it a nonadjacent digit set (NADS). We present an algorithm to determine if {0,1,x} is a NADS; and if it is, we present an algorithm to efficiently determine the nonadjacent representation of any positive integer. We also present some necessary and sufficient conditions for {0,1,x} to be a NADS. These conditions are used to exhibit infinite families of integers x such that {0,1,x} is a NADS, as well as infinite families of x such that {0,1,x} is not a NADS.
Chapter PDF
References
Booth, A.D.: A Signed Binary Multiplication Technique. Quarterly Journal of Mechanics and Applied Mathematics 4, 236–240 (1951)
Gordon, D.M.: A Survey of Fast Exponentiation Methods. Journal of Algorithms 27, 129–146 (1998)
Jedwab, J., Mitchell, C.J.: Minimum Weight Modified Signed-Digit Representations and Fast Exponentiation. Electronic Letters 25, 1171–1172 (1989)
Matula, D.W.: Basic Digit Sets for Radix Representation. Journal of the Association for Computing Machinery 29, 1131–1143 (1982)
Menezes, J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996)
Morain, F., Olivos, J.: Speeding up the Computations on an Elliptic Curve using Addition-Subtraction Chains. RAIRO Theoretical Informatics and Applications 24, 531–543 (1990)
Reitwiesner, G.W.: Binary Arithmetic. In: Advances in Computers, vol. 1, pp. 231–308. Academic Press, London (1960)
Solinas, J.A.: Low-Weight Binary Representations for Pairs of Integers. Technical Report CORR 2001-41, Centre for Applied Cryptographic Research, Available from http://www.cacr.math.uwaterloo.ca/-techreports/2001/corr2001-41.ps
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)
Solinas, J.A.: Efficient arithmetic on Koblitz curves. Designs, Codes and Cryptography 19, 195–249 (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Muir, J.A., Stinson, D.R. (2004). Alternative Digit Sets for Nonadjacent Representations. In: Matsui, M., Zuccherato, R.J. (eds) Selected Areas in Cryptography. SAC 2003. Lecture Notes in Computer Science, vol 3006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24654-1_22
Download citation
DOI: https://doi.org/10.1007/978-3-540-24654-1_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21370-3
Online ISBN: 978-3-540-24654-1
eBook Packages: Springer Book Archive