Existence and optimality of w-non-adjacent forms with an algebraic integer base
- 87 Downloads
We consider digit expansions in lattices with endomorphisms acting as base. We focus on the w-non-adjacent form (w-NAF), where each block of w consecutive digits contains at most one non-zero digit. We prove that for sufficiently large w and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a w-NAF.
If the eigenvalues of the endomorphism are large enough and w is sufficiently large, then the w-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.
Key words and phrasesτ-adic expansion w-non-adjacent form redundant digit set lattice existence hyperelliptic curve cryptography Koblitz curve Frobenius endomorphism scalar multiplication Hamming weight optimality minimal expansion
Mathematics Subject Classification11A63 11H06 11R04 94A60
Unable to display preview. Download preview PDF.
- R. M. Avanzi, A note on the signed sliding window integer recoding and a left-to-right analogue, in H. Handschuh and A. Hasan (Eds.), Selected Areas in Cryptography: 11th International Workshop, SAC 2004 (Waterloo, Canada, August 9–10, 2004), Revised Selected Papers, Lecture Notes in Comput. Sci. 3357, Springer-Verlag (Berlin, 2004), pp. 130–143. CrossRefGoogle Scholar
- C. Heuberger and D. Krenn, Optimality of the width-w non-adjacent form: General characterisation and the case of imaginary quadratic bases, to appear in J. Théor. Nombres Bordeaux (2013), earlier version available at arXiv:1110.0966v1 [math.NT] (2011).
- C. Heuberger and D. Krenn, Analysis of width-w non-adjacent forms to imaginary quadratic bases, to appear in J. Number Theory (2012), earlier version available at arXiv:1009.0488v2 [math.NT].
- J. A. Solinas, An improved algorithm for arithmetic on a family of elliptic curves, in B. S. Kaliski, Jr. (Ed.), Advances in Cryptology – CRYPTO ’97. 17th Annual International Cryptology Conference (Santa Barbara, CA, USA, August 17–21, 1997), Proceedings, Lecture Notes in Comput. Sci. 1294, Springer (Berlin, 1997), pp. 357–371. CrossRefGoogle Scholar