On Τ-adic representations of integers

Authors:
Nevine Maurice Ebeid;M. Anwar Hasan
Affiliations:
Department of Electrical and Computer Engineering, Centre for Applied Cryptographic Research, University of Waterloo, Waterloo, Canada N2L 3G1;Department of Electrical and Computer Engineering, Centre for Applied Cryptographic Research, University of Waterloo, Waterloo, Canada N2L 3G1
Venue:
Designs, Codes and Cryptography
Year:
2007

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Abstract

Elliptic curve cryptosystems (ECCs) have become increasingly popular due to their efficiency and the small size of the keys they use. Particularly, the anomalous curves introduced by Koblitz allow a complex representation of the keys, denoted 驴NAF, that make the computations over these curves more efficient. In this article, we propose an efficient method for randomizing a 驴NAF to produce different equivalent representations of the same key to the same complex base 驴. We prove that the average Hamming density of the resulting representations is 0.5. We identify the pattern of the 驴NAFs yielding the maximum number of representations and the formula governing this number. We also present deterministic methods to compute the average and the exact number of possible representations of a 驴NAF.