Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
Journal of Cryptology
A survey of fast exponentiation methods
Journal of Algorithms
Optimal Left-to-Right Binary Signed-Digit Recoding
IEEE Transactions on Computers - Special issue on computer arithmetic
Efficient elliptic curve exponentiation
ICICS '97 Proceedings of the First International Conference on Information and Communication Security
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
New Minimal Modified Radix-r Representation with Applications to Smart Cards
PKC '02 Proceedings of the 5th International Workshop on Practice and Theory in Public Key Cryptosystems: Public Key Cryptography
Efficient Subgroup Exponentiation in Quadratic and Sixth Degree Extensions
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
Distribution results for low-weight binary representations for pairs of integers
Theoretical Computer Science - Combinatorics of the discrete plane and tilings
Analysis of linear combination algorithms in cryptography
ACM Transactions on Algorithms (TALG)
The Complexity of Certain Multi-Exponentiation Techniques in Cryptography
Journal of Cryptology
Analysis of Fractional Window Recoding Methods and Their Application to Elliptic Curve Cryptosystems
IEEE Transactions on Computers
Theoretical Computer Science
Faster Multi-exponentiation through Caching: Accelerating (EC)DSA Signature Verification
SCN '08 Proceedings of the 6th international conference on Security and Cryptography for Networks
New left-to-right minimal weight signed-digit radix-r representation
Computers and Electrical Engineering
A New Upper Bound for the Minimal Density of Joint Representations in Elliptic Curve Cryptosystems
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Unbalanced digit sets and the closest choice strategy for minimal weight integer representations
Designs, Codes and Cryptography
Jacobi Quartic Curves Revisited
ACISP '09 Proceedings of the 14th Australasian Conference on Information Security and Privacy
Delaying and merging operations in scalar multiplication: applications to curve-based cryptosystems
SAC'06 Proceedings of the 13th international conference on Selected areas in cryptography
On redundant T-adic expansions and non-adjacent digit sets
SAC'06 Proceedings of the 13th international conference on Selected areas in cryptography
Secure signed radix-r recoding methods for constrained-embedded devices
ISPEC'07 Proceedings of the 3rd international conference on Information security practice and experience
New left-to-right radix-r signed-digit recoding algorithm for pairing-based cryptosystems
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
Affine precomputation with sole inversion in elliptic curve cryptography
ACISP'07 Proceedings of the 12th Australasian conference on Information security and privacy
Efficient techniques for high-speed elliptic curve cryptography
CHES'10 Proceedings of the 12th international conference on Cryptographic hardware and embedded systems
An advanced method for joint scalar multiplications on memory constraint devices
ESAS'05 Proceedings of the Second European conference on Security and Privacy in Ad-Hoc and Sensor Networks
A note on signed binary window algorithm for elliptic curve cryptosystems
CANS'05 Proceedings of the 4th international conference on Cryptology and Network Security
New minimal weight representations for left-to-right window methods
CT-RSA'05 Proceedings of the 2005 international conference on Topics in Cryptology
Fractional windows revisited: improved signed-digit representations for efficient exponentiation
ICISC'04 Proceedings of the 7th international conference on Information Security and Cryptology
SPA resistant left-to-right integer recodings
SAC'05 Proceedings of the 12th international conference on Selected Areas in Cryptography
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Addition-subtraction-chains obtained from signed digit recodings of integers are a common tool for computing multiples of random elements of a group where the computation of inverses is a fast operation. Cohen and Solinas independently described one such recoding, the ω-NAF. For scalars of the size commonly used in cryptographic applications, it leads to the current scalar multiplication algorithm of choice. However, we could find no formal proof of its optimality in the literature. This recoding is computed right-to-left. We solve two open questions regarding the ω-NAF. We first prove that the ω-NAF is a redundant radix-2 recoding of smallest weight among all those with integral coefficients smaller in absolute value than 2ω−1. Secondly, we introduce a left-to-right recoding with the same digit set as the ω-NAF, generalizing previous results. We also prove that the two recodings have the same (optimal) weight. Finally, we sketch how to prove similar results for other recodings.