A course in computational algebraic number theory
A course in computational algebraic number theory
A survey of fast exponentiation methods
Journal of Algorithms
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
SAC '02 Revised Papers from the 9th Annual International Workshop on Selected Areas in Cryptography
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
An Alternate Decomposition of an Integer for Faster Point Multiplication on Certain Elliptic Curves
PKC '02 Proceedings of the 5th International Workshop on Practice and Theory in Public Key Cryptosystems: Public Key Cryptography
An Improved Algorithm for uP + vQ on a Family of Elliptic Curves
IPDPS '05 Proceedings of the 19th IEEE International Parallel and Distributed Processing Symposium (IPDPS'05) - Workshop 17 - Volume 18
FPGA Design of Self-certified Signature Verification on Koblitz Curves
CHES '07 Proceedings of the 9th international workshop on Cryptographic Hardware and Embedded Systems
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
Left-to-Right signed-bit τ-adic representations of n integers
ICICS'06 Proceedings of the 8th international conference on Information and Communications Security
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Mycrypt'05 Proceedings of the 1st international conference on Progress in Cryptology in Malaysia
Efficiently computable endomorphisms for hyperelliptic curves
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
New families of hyperelliptic curves with efficient gallant-lambert-vanstone method
ICISC'04 Proceedings of the 7th international conference on Information Security and Cryptology
Efficient scalar multiplication by isogeny decompositions
PKC'06 Proceedings of the 9th international conference on Theory and Practice of Public-Key Cryptography
A DPA countermeasure by randomized frobenius decomposition
WISA'05 Proceedings of the 6th international conference on Information Security Applications
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In most algorithms involving elliptic curves, the most expensive part consists in computing multiples of points. This paper investigates how to extend the τ -adic expansion from Koblitz curves to a larger class of curves defined over a prime field having an efficiently-computable endomorphism φ in order to perform an efficient point multiplication with efficiency similar to Solinas' approach presented at CRYPTO '97. Furthermore, many elliptic curve cryptosystems require the computation of k0P + k1Q. Following the work of Solinas on the Joint Sparse Form, we introduce the notion of φ-Joint Sparse Form which combines the advantages of a φ-expansion with the additional speedup of the Joint Sparse Form. We also present an efficient algorithm to obtain the φ-Joint Sparse Form. Then, the double exponentiation can be done using the φ endomorphism instead of doubling, resulting in an average of l applications of φ and l/2 additions, where l is the size of the ki's. This results in an important speed-up when the computation of φ is particularly effective, as in the case of Koblitz curves.