Efficient Arithmetic on Koblitz Curves
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Elliptic curves in cryptography
Elliptic curves in cryptography
Optimal Left-to-Right Binary Signed-Digit Recoding
IEEE Transactions on Computers - Special issue on computer arithmetic
Signed Digit Representations of Minimal Hamming Weight
IEEE Transactions on Computers
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
Elliptic Scalar Multiplication Using Point Halving
ASIACRYPT '99 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
On the Performance of Signature Schemes Based on Elliptic Curves
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
Guide to Elliptic Curve Cryptography
Guide to Elliptic Curve Cryptography
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Efficient implementations of cryptosystems are important in order to conserve resources, memory, power, etc., which will enable resource-limited devices to compute necessary cryptographic operations. One technique that successfully reduces the number of necessary operations is the use of a signed digit representation for the key, because it reduces the nonzero density of the representation. One such signed digit representation is the non-adjacent form or NAF. Moreover, one can make more reductions in the number of nonzero symbols of the key by expressing the key with a w-ary NAF or wNAF form. A drawback is that one needs to parse the key twice, once to construct the wNAF representation and the second time to perform the necessary cryptographic operation. At Crypto 2004 [10], Okeya et. al. introduced a w-ary representation wMOF, which possess the same nonzero density as wNAF, as well as an algorithm that computes wNOF in a left-to-right manner utilizing very little memory ("memory-less"). At that time, the authors noted that a left-to-right "memory-less" algorithm that computes wNAF is an open problem. In this work, we define wNAF, a generalization of wNAF. Further, we construct a left-to-right "memory-less" algorithm that computes the w-ary wNAF representation of a key and demonstrate that wNAF is as efficient as wNAF. Our work will demonstrate that the left-to-right wNAF recoding algorithm closely resembles the right-to-left wNAF recoding algorithm.