A survey of fast exponentiation methods
Journal of Algorithms
Exponentiation Using Division Chains
IEEE Transactions on Computers
MIST: An Efficient, Randomized Exponentiation Algorithm for Resisting Power Analysis
CT-RSA '02 Proceedings of the The Cryptographer's Track at the RSA Conference on Topics in Cryptology
Some Security Aspects of the M IST Randomized Exponentiation Algorithm
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
Exponentiation using Division Chains
ARITH '97 Proceedings of the 13th Symposium on Computer Arithmetic (ARITH '97)
Theory and applications for a double-base number system
ARITH '97 Proceedings of the 13th Symposium on Computer Arithmetic (ARITH '97)
Trading Inversions for Multiplications in Elliptic Curve Cryptography
Designs, Codes and Cryptography
Fast Multibase Methods and Other Several Optimizations for Elliptic Curve Scalar Multiplication
Irvine Proceedings of the 12th International Conference on Practice and Theory in Public Key Cryptography: PKC '09
The Jacobi model of an elliptic curve and side-channel analysis
AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
PKC'08 Proceedings of the Practice and theory in public key cryptography, 11th international conference on Public key cryptography
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
Efficient scalar multiplication by isogeny decompositions
PKC'06 Proceedings of the 9th international conference on Theory and Practice of Public-Key Cryptography
ISC'07 Proceedings of the 10th international conference on Information Security
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There have been many recent developments in formulae for efficient composite elliptic curve operations of the form dP+Q for a small integer d and points P and Q where the underlying field is a prime field. To make best use of these in a scalar multiplication kP, it is necessary to generate an efficient "division chain" for the scalar where divisions of k are by the values of d available through composite operations. An algorithm-generating algorithm for this is presented that takes into account the different costs of using various representations for curve points. This extends the applicability of methods presented by Longa & Gebotys at PKC 2009 to using specific characteristics of the target device. It also enables the transfer of some scalar recoding computation details to design time. An improved cost function also provides better evaluation of alternatives in the relevant addition chain. One result of these more general and improved methods includes a slight increase over the scalar multiplication speeds reported at PKC. Furthermore, by the straightforward removal of rules for unusual cases, some particularly concise yet efficient presentations can be given for algorithms in the target device.