doi:10.1016/S0164-1212(98)00007-7 
Copyright © 1998 Elsevier Science Inc. All rights reserved
An adaptive exponentiation method
Der-Chyuan Loua, * and Chin-Chen Chang1, b
a Department of Electrical Engineering, Chung Cheng Institute of Technology, Tahsi, Taoyuan 335, Taiwan, ROC
b Institute of Computer Science and Information Engineering, National Chung Cheng University, Chiayi 621, Taiwan, ROC
Received 16 September 1996;
revised 11 October 1996;
accepted 11 December 1996.
Available online 30 November 1998.
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Abstract
Exponentiation of large positive integer with a 512-bit exponent is the basis of several well-known cryptographic algorithms. In this paper, an adaptive method for improving the performance of the m-ary method is proposed and analyzed. Due to the efficient utilization of partial results, it is useful for systems with varied exponent and base. This method is based on two ideas. Firstly, for base 
, a few of the exponentiations with smaller exponents are precomputed on-line, that is, 
are precomputed, where 
is an optimization parameter. Secondly, a number of used partial results will be determined and stored in a look-up table during the computation. Assume that squaring is free as compared with multiplication, depending on the numbers of precomputations and partial results, the proposed method on average gives a 26–40% time reduction as compared with the 
-ary method. On the other hand, it does require little temporary storage for the used partial results.
Author Keywords: Public key cryptography; Exponentiation; The binary method; The
-ary method
Fig. 1. The general structure of the tree for (i.e., ).
Table 1. Performance comparisons of various exponentiation methods (for the average case of a random 512-bit exponent)
Table 2. The average number of multiplications needed in the -ary method (the bold numbers indicate the minimal values)
Table 3. The average number of multiplications needed in the new method (the bold numbers indicate the minimal values)
Table 4. The speedup of the new method under the condition the squaring is free as compared with multiplication
Table 5. The average number of multiplications and squarings needed in the -ary method (the bold numbers indicate the minimal values)
Table 6. The average number of multiplications and squarings needed in the new method (the bold numbers indicate the minimal values)
Table 7. The speedup of the new method under the condition that squaring takes the same time as multiplication
Abstract
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