Analysis of the Width-w Non-Adjacent Form in Conjunction with Hyperelliptic Curve Cryptography and with Lattices
Abstract
We analyse the number of occurrences of a fixed non-zero digit in the width-w non-adjacent forms of all elements of a lattice in some region (e.g. a ball). Our result is an asymptotic formula, where its main term coincides with the full block length analysis. In its second order term a periodic fluctuation is exhibited. The proof follows Delange's method. This result in a general lattice set-up is then used for numeral systems with an algebraic integer as base. Those come from efficient scalar multiplication methods (Frobenius-and-add methods) in hyperelliptic curves cryptography, and our result is needed for analysing the running time of such algorithms.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2012
- DOI:
- 10.48550/arXiv.1209.0618
- arXiv:
- arXiv:1209.0618
- Bibcode:
- 2012arXiv1209.0618K
- Keywords:
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- Mathematics - Number Theory;
- 11A63;
- 11H99;
- 11R21;
- 28A80;
- 94A60
- E-Print:
- arXiv admin note: substantial text overlap with arXiv:1009.0488