Advances in Applied Mathematics
Efficient Arithmetic on Koblitz Curves
Designs, Codes and Cryptography - Special issue on towards a quarter-century of public key cryptography
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
Efficient Algorithms for Pairing-Based Cryptosystems
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
An Elliptic Curve Implementation of the Finite Field Digital Signature Algorithm
CRYPTO '98 Proceedings of the 18th 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
An Algorithm for the nt Pairing Calculation in Characteristic Three and its Hardware Implementation
ARITH '07 Proceedings of the 18th IEEE Symposium on Computer Arithmetic
Software Implementation of Arithmetic in
WAIFI '07 Proceedings of the 1st international workshop on Arithmetic of Finite Fields
A New Direct Anonymous Attestation Scheme from Bilinear Maps
Trust '08 Proceedings of the 1st international conference on Trusted Computing and Trust in Information Technologies: Trusted Computing - Challenges and Applications
Pairing '08 Proceedings of the 2nd international conference on Pairing-Based Cryptography
Multi-core Implementation of the Tate Pairing over Supersingular Elliptic Curves
CANS '09 Proceedings of the 8th International Conference on Cryptology and Network Security
A Taxonomy of Pairing-Friendly Elliptic Curves
Journal of Cryptology
On redundant T-adic expansions and non-adjacent digit sets
SAC'06 Proceedings of the 13th international conference on Selected areas in cryptography
Designs, Codes and Cryptography
Extending scalar multiplication using double bases
ASIACRYPT'06 Proceedings of the 12th international conference on Theory and Application of Cryptology and Information Security
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We describe new algorithms for performing scalar multiplication on supersingular elliptic curves in characteristic three. These curves can be used in pairing-based cryptography. Since in pairing-based protocols besides pairing computations also scalar multiplications are required, and the performance of the latter is not negligible, improving it is clearly important as well. The techniques presented here bring noticeable speed ups (up to 30% for methods using a variable amount of memory and up to 46.7% for methods with a small, fixed memory usage), while at the same time bringing substantial memory reductions - factors like 3 to 8 are not uncommon. The starting point for our methods is a structure theorem for unit groups of residue classes of a quadratic order associated to the Frobenius endomorphism of the considered curves. This allows us to define new digit sets whose elements are products of powers of certain generators of said groups. There are of course several choices for these generators: we chose generators associated to endomorphisms for which we could find efficient explicit formulae in a suitable coordinate system. A multiple-base-like scalar multiplication algorithm making use of these digits and these formulae brings the claimed speed up.