A cryptographic library for the Motorola DSP56000
EUROCRYPT '90 Proceedings of the workshop on the theory and application of cryptographic techniques on Advances in cryptology
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A survey of fast exponentiation methods
Journal of Algorithms
A common-multiplicand method to the Montgomery algorithm for speeding up exponentiation
Information Processing Letters
An improved Montgomery's algorithm for high-speed RSA public-key cryptosystem
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
Optimal Left-to-Right Binary Signed-Digit Recoding
IEEE Transactions on Computers - Special issue on computer arithmetic
A method for obtaining digital signatures and public-key cryptosystems
Communications of the ACM
A Fast Algorithm for Multiplicative Inversion in GF(2m) Using Normal Basis
IEEE Transactions on Computers
Computer Arithmetic Algorithms
Computer Arithmetic Algorithms
Signed Digit Representations of Minimal Hamming Weight
IEEE Transactions on Computers
Fast Algorithms for Common-Multiplicand Multiplication and Exponentiation by Performing Complements
AINA '03 Proceedings of the 17th International Conference on Advanced Information Networking and Applications
A Scalable Architecture for Modular Multiplication Based on Montgomery's Algorithm
IEEE Transactions on Computers
Fast exponentiation based on common-multiplicand-multiplication and minimal-signed-digit techniques
International Journal of Computer Mathematics
GPU-Accelerated Montgomery Exponentiation
ICCS '07 Proceedings of the 7th international conference on Computational Science, Part I: ICCS 2007
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Efficient computation of the modular exponentiations is veryimportant and useful for public-key cryptosystems. In this paper,an efficient parallel binary exponentiation algorithm is proposedwhich based on the Montgomery multiplication algorithm, thesigned-digit-folding (SDF) and common-multiplicand-multiplicand(CMM) techniques. By using the CMM technique of computing thecommon part from two modular multiplications, the same common partin two modular multiplications can be computed once rather twice,we can thus improve the efficiency of the binary exponentiationalgorithm by decreasing the number of modular multiplications. Bydividing the bit pattern of the minimal-signed-digit recodingexponent into three equal length parts and using the technique ofrecording the common parts in the folded substrings, the proposedSDF-CMM algorithm can improve the efficiency of the binaryalgorithm, thus can further decrease the computational complexityof modular exponentiation. Furthermore, by using the proposedparallel SDF-CMM Montgomery binary exponentiation algorithm, onaverage the total number of single-precision multiplications can bereduced by about 61.3% and 74.1% as compared with Chang-Kuo-Lin'sCMM modular exponentiation algorithm and Ha-Moon's CMM Montgomerymodular exponentiation algorithm, respectively.