Advances in Applied Mathematics
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Improving the arithmetic of elliptic curves in the Jacobi model
Information Processing Letters
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Endomorphisms for Faster Elliptic Curve Cryptography on a Large Class of Curves
EUROCRYPT '09 Proceedings of the 28th Annual International Conference on Advances in Cryptology: the Theory and Applications of Cryptographic Techniques
Novel Precomputation Schemes for Elliptic Curve Cryptosystems
ACNS '09 Proceedings of the 7th International Conference on Applied Cryptography and Network Security
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AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
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ICISC'02 Proceedings of the 5th international conference on Information security and cryptology
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CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
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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
AFRICACRYPT'08 Proceedings of the Cryptology in Africa 1st international conference on Progress in cryptology
Faster group operations on elliptic curves
AISC '09 Proceedings of the Seventh Australasian Conference on Information Security - Volume 98
A note on the signed sliding window integer recoding and a left-to-right analogue
SAC'04 Proceedings of the 11th international conference on Selected Areas in Cryptography
Tate pairing computation on jacobi's elliptic curves
Pairing'12 Proceedings of the 5th international conference on Pairing-Based Cryptography
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This paper provides new results about efficient arithmetic on Jacobi quartic form elliptic curves, y 2 = d x 4 + 2 a x 2 + 1. With recent bandwidth-efficient proposals, the arithmetic on Jacobi quartic curves became solidly faster than that of Weierstrass curves. These proposals use up to 7 coordinates to represent a single point. However, fast scalar multiplication algorithms based on windowing techniques, precompute and store several points which require more space than what it takes with 3 coordinates. Also note that some of these proposals require d = 1 for full speed. Unfortunately, elliptic curves having 2-times-a-prime number of points, cannot be written in Jacobi quartic form if d = 1. Even worse the contemporary formulae may fail to output correct coordinates for some inputs. This paper provides improved speeds using fewer coordinates without causing the above mentioned problems. For instance, our proposed point doubling algorithm takes only 2 multiplications, 5 squarings, and no multiplication with curve constants when d is arbitrary and a = ±1/2.