A survey of fast exponentiation methods
Journal of Algorithms
Basic digit sets for radix representation
Journal of the ACM (JACM)
Efficient Arithmetic on Koblitz Curves
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Optimal Left-to-Right Binary Signed-Digit Recoding
IEEE Transactions on Computers - Special issue on computer arithmetic
Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method
CRYPTO '92 Proceedings of the 12th Annual International Cryptology Conference on Advances in Cryptology
An Improved Algorithm for Arithmetic on a Family of Elliptic Curves
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
Compact Encoding of Non-adjacent Forms with Applications to Elliptic Curve Cryptography
PKC '01 Proceedings of the 4th International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
On the Performance of Signature Schemes Based on Elliptic Curves
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
Field inversion and point halving revisited
IEEE Transactions on Computers
Designs, Codes and Cryptography
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From several decades, non-adjacent form (NAF) representations for integers have been extensively studied as an alternative to the usual binary number system where digits are in {0,1}. In cryptography, the non-adjacent digit set (NADS) {–1,0,1} is used for optimization of arithmetic operations in elliptic curves. At SAC 2003, Muir and Stinson published new results on alternative digit sets: they proposed infinite families of integers x such that {0,1,x} is a NADS as well as infinite families of integers x such that {0,1,x} is not a NADS, so called a NON-NADS. Muir and Stinson also provided an algorithm that determines whether x leads to a NADS by checking if every integer $n \epsilon [0, \lfloor \frac{-x}{3} \rfloor]$ has a {0,1,x}-NAF. In this paper, we extend these results by providing generators of NON-NADS infinite families. Furthermore, we reduce the search bound from $\lfloor \frac{-x}{3} \rfloor$ to $\lfloor \frac{-x}{12} \rfloor$. We introduce the notion of worst NON-NADS and give the complete characterization of such sets. Beyond the theoretical results, our contribution also aims at exploring some algorithmic aspects. We supply a much more efficient algorithm than those proposed by Muir and Stinson, which takes only 343 seconds to compute all x’s from 0 to –107 such that {0,1,x} is a NADS.