# Modification Of Bos & Coster's Heuristics in Search of a Shorter Addition Chain for Faster Exponentiation.

December 2011

## Date of Award

Winter 12-12-2012

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Master's Dissertation

## Degree Name

Master of Technology

Computer Science

## Department

Applied Statistics Unit (ASU-Kolkata)

## Supervisor

Sarkar, Palash (ASU-Kolkata; ISI)

## Abstract (Summary of the Work)

The basic question that necessitates a study of addition chains is: What is the fewest number of multiplications necessary to compute g r , given that the only operation permitted is multiplying two already-computed powers? This is equivalent to the question: What is the length of the shortest addition chain for r?An addition chain for r is a list of positive integers a1 = 1, a2, . . . , al = r, such that, for each i > 1, there is some j and k with 1 â‰¤ j â‰¤ k < i and ai = aj + ak.A short addition chain for r gives a fast algorithm for computing g r : compute g a1 , ga2 , . . . , galâˆ’1 , gr . Let l(r) be the length of the shortest addition chain for r. The exact value of l(r) is known only for relatively small values of r. It is known that, for large r, l(r) = log r + (1+o(1))log r/log logr.The lower bound was shown in [Erdos60] using a counting argument and the upper bound is given by the m-ary method discussed in 2.1. In 2, few basic methods and the double base number system used for fast exponentiation are discussed. In 3, outline of few algorithms for finding an addition chain containing a number n has been provided. 4 constitutes the three different algorithms we have designed as modifications to the Makesequence algorithm of Bos and Coster [BosCoster90]. In 5, in the first part we discuss the results obtained from running the C programs for Brauerâ€™s and Yaoâ€™s algorithm for various n and k. In the later part we compare our algorithms with the original Bos and Coster method with respect to a test examp

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843091

## Control Number

ISI-DISS-2011-285