The implementation of elliptic curve cryptosystems
AUSCRYPT '90 Proceedings of the international conference on cryptology on Advances in cryptology
Theory and Applications of the Double-Base Number System
IEEE Transactions on Computers
Efficient Arithmetic on Koblitz Curves
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
Efficient elliptic curve exponentiation
ICICS '97 Proceedings of the First International Conference on Information and Communication Security
Faster Point Multiplication on Elliptic Curves with Efficient Endomorphisms
CRYPTO '01 Proceedings of the 21st 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
Efficient Elliptic Curve Exponentiation Using Mixed Coordinates
ASIACRYPT '98 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Guide to Elliptic Curve Cryptography
Guide to Elliptic Curve Cryptography
Trading Inversions for Multiplications in Elliptic Curve Cryptography
Designs, Codes and Cryptography
Fast point multiplication on elliptic curves through isogenies
AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
Improved algorithms for efficient arithmetic on elliptic curves using fast endomorphisms
EUROCRYPT'03 Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques
Do all elliptic curves of the same order have the same difficulty of discrete log?
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
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
Finite Fields and Their Applications
A Tree-Based Approach for Computing Double-Base Chains
ACISP '08 Proceedings of the 13th Australasian conference on Information Security and Privacy
Twisted Edwards Curves Revisited
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Generalized Scalar Multiplication Secure against SPA, DPA, and RPA
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Elliptic Curve Scalar Multiplication Combining Yao's Algorithm and Double Bases
CHES '09 Proceedings of the 11th International Workshop on Cryptographic Hardware and Embedded Systems
On the number of distinct elliptic curves in some families
Designs, Codes and Cryptography
Generalized MMM-algorithm secure against SPA, DPA, and RPA
ICISC'07 Proceedings of the 10th international conference on Information security and cryptology
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
Optimizing double-base elliptic-curve single-scalar multiplication
INDOCRYPT'07 Proceedings of the cryptology 8th international conference on Progress in cryptology
Faster addition and doubling on elliptic curves
ASIACRYPT'07 Proceedings of the Advances in Crypotology 13th international conference on Theory and application of cryptology and information security
PKC'08 Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography
Faster group operations on elliptic curves
AISC '09 Proceedings of the Seventh Australasian Conference on Information Security - Volume 98
Fast scalar multiplication for ECC over GF(p) using division chains
WISA'10 Proceedings of the 11th international conference on Information security applications
Group law computations on jacobians of hyperelliptic curves
SAC'11 Proceedings of the 18th international conference on Selected Areas in Cryptography
On various families of twisted jacobi quartics
SAC'11 Proceedings of the 18th international conference on Selected Areas in Cryptography
ISC'07 Proceedings of the 10th international conference on Information Security
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On an elliptic curve, the degree of an isogeny corresponds essentially to the degrees of the polynomial expressions involved in its application. The multiplication–by–ℓ map [ℓ] has degree ℓ2, therefore the complexity to directly evaluate [ℓ](p) is O(ℓ2). For a small prime ℓ (= 2, 3) such that the additive binary representation provides no better performance, this represents the true cost of application of scalar multiplication. If an elliptic curve admits an isogeny ϕ of degree ℓ then the costs of computing ϕ(P) should in contrast be O(ℓ) field operations. Since we then have a product expression [ℓ]=$\hat{\varphi}\varphi$, the existence of an ℓ-isogeny ϕ on an elliptic curve yields a theoretical improvement from O(ℓ2) to O(ℓ) field operations for the evaluation of [ℓ](p) by naïve application of the defining polynomials. In this work we investigate actual improvements for small ℓ of this asymptotic complexity. For this purpose, we describe the general construction of families of curves with a suitable decomposition [ℓ]=$\hat{\varphi}\varphi$, and provide explicit examples of such a family of curves with simple decomposition for [3]. Finally we derive a new tripling algorithm to find complexity improvements to triplication on a curve in certain projective coordinate systems, then combine this new operation to non-adjacent forms for ℓ-adic expansions in order to obtain an improved strategy for scalar multiplication on elliptic curves.