An algorithm for modular exponentiation
Information Processing Letters
Efficient Arithmetic on Koblitz Curves
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Elliptic Curve Public Key Cryptosystems
Elliptic Curve Public Key Cryptosystems
Fast Generation of Pairs (k, [k]P) for Koblitz Elliptic Curves
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
Efficient Algorithms for Pairing-Based Cryptosystems
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
CM-Curves with Good Cryptographic Properties
CRYPTO '91 Proceedings of the 11th 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
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 redundant T-adic expansions and non-adjacent digit sets
SAC'06 Proceedings of the 13th international conference on Selected areas in cryptography
Scalar multiplication on koblitz curves using double bases
VIETCRYPT'06 Proceedings of the First international conference on Cryptology in Vietnam
Efficient and secure elliptic curve point multiplication using double-base chains
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
An analysis of double base number systems and a sublinear scalar multiplication algorithm
Mycrypt'05 Proceedings of the 1st international conference on Progress in Cryptology in Malaysia
FPGA implementation of point multiplication on koblitz curves using kleinian integers
CHES'06 Proceedings of the 8th international conference on Cryptographic Hardware and Embedded Systems
Short memory scalar multiplication on koblitz curves
CHES'05 Proceedings of the 7th international conference on Cryptographic hardware and embedded systems
SAC'05 Proceedings of the 12th international conference on Selected Areas in Cryptography
Field inversion and point halving revisited
IEEE Transactions on Computers
A Tree-Based Approach for Computing Double-Base Chains
ACISP '08 Proceedings of the 13th Australasian conference on Information Security and Privacy
Double-Base Number System for Multi-scalar Multiplications
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
On redundant T-adic expansions and non-adjacent digit sets
SAC'06 Proceedings of the 13th international conference on Selected areas in cryptography
New formulae for efficient elliptic curve arithmetic
INDOCRYPT'07 Proceedings of the cryptology 8th international conference on Progress in cryptology
A graph theoretic analysis of double base number systems
INDOCRYPT'07 Proceedings of the cryptology 8th international conference on Progress in cryptology
Designs, Codes and Cryptography
Faster and lower memory scalar multiplication on supersingular curves in characteristic three
PKC'11 Proceedings of the 14th international conference on Practice and theory in public key cryptography conference on Public key cryptography
Scalar multiplication on koblitz curves using double bases
VIETCRYPT'06 Proceedings of the First international conference on Cryptology in Vietnam
Extended double-base number system with applications to elliptic curve cryptography
INDOCRYPT'06 Proceedings of the 7th international conference on Cryptology in India
Efficient arithmetic on hessian curves
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
Sublinear scalar multiplication on hyperelliptic koblitz curves
SAC'11 Proceedings of the 18th international conference on Selected Areas in Cryptography
ISC'07 Proceedings of the 10th international conference on Information Security
A new algorithm for computing triple-base number system
ACM SIGARCH Computer Architecture News
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It has been recently acknowledged [4,6,9] that the use of double bases representations of scalars n, that is an expression of the form n = ∑e, s, t (–1)eAsBt can speed up significantly scalar multiplication on those elliptic curves where multiplication by one base (say B) is fast. This is the case in particular of Koblitz curves and supersingular curves, where scalar multiplication can now be achieved in o(logn) curve additions. Previous literature dealt basically with supersingular curves (in characteristic 3, although the methods can be easily extended to arbitrary characteristic), where A,B ∈ℕ. Only [4] attempted to provide a similar method for Koblitz curves, where at least one base must be non-real, although their method does not seem practical for cryptographic sizes (it is only asymptotic), since the constants involved are too large. We provide here a unifying theory by proposing an alternate recoding algorithm which works in all cases with optimal constants. Furthermore, it can also solve the until now untreatable case where both A and B are non-real. The resulting scalar multiplication method is then compared to standard methods for Koblitz curves. It runs in less than logn/loglogn elliptic curve additions, and is faster than any given method with similar storage requirements already on the curve K-163, with larger improvements as the size of the curve increases, surpassing 50% with respect to the τ-NAF for the curves K-409 and K-571. With respect of windowed methods, that can approach our speed but require O(log(n)/loglog(n)) precomputations for optimal parameters, we offer the advantage of a fixed, small memory footprint, as we need storage for at most two additional points.