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Radix-r Non-Adjacent Form and Its Application to Pairing-Based Cryptosystem Tsuyoshi TAKAGI David REIS, Jr. Sung-Ming YEN Bo-Ching WU Publication IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences Vol.E89-A No.1 pp.115-123 Publication Date: 2006/01/01 Online ISSN: 1745-1337 DOI: 10.1093/ietfec/e89-a.1.115 Print ISSN: 0916-8508 Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security) Category: Elliptic Curve Cryptography Keyword: non-adjacent form, radix-r representation, signed window method, elliptic curve cryptosystem, pairing based cryptosystem, Full Text: PDF(209.6KB)>>
Summary: Recently, the radix-3 representation of integers is used for the efficient implementation of pairing based cryptosystems. In this paper, we propose non-adjacent form of radix-r representation (rNAF) and efficient algorithms for generating rNAF. The number of non-trivial digits is (r-2)(r+1)/2 and its average density of non-zero digit is asymptotically (r-1)/(2r-1). For r=3, the non-trivial digits are {  2, 4} and the non-zero density is 0.4. We then investigate the width-w version of rNAF for the general radix-r representation, which is a natural extension of the width-w NAF. Finally we compare the proposed algorithms with the generalized NAF (gNAF) discussed by Joye and Yen. The proposed scheme requires a larger table but its non-zero density is smaller even for large radix. We explain that gNAF is a simple degeneration of rNAF--we can consider that rNAF is a canonical form for the radix-r representation. Therefore, rNAF is a good alternative to gNAF. | ||||||||||
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