ExponentiationFor exponentiation, the MPI library uses a simple and fairly standardsquare-and-multiply method.  The algorithm is this:Input:	a, bOutput: a ** b	s = 1	while(b != 0)	  if(b is odd)	    s = s * a	  endif	  b = b / 2	  x = x * x	endwhile	return sThe modular exponentiation is done the same way, except replacing:	s = s * awith	s = (s * a) mod mand replacing	x = x * xwith	x = (x * x) mod mHere is a sample exponentiation using the MPI library, as compared tothe same problem solved by the Unix 'bc' program on my system:Computation of 2,381,283 ** 235'bc' says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says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iff says:% diff bc.txt mp.txt%------------------------------------------------------------------The contents of this file are subject to the Mozilla PublicLicense Version 1.1 (the "License"); you may not use this fileexcept in compliance with the License. You may obtain a copy ofthe License at http://www.mozilla.org/MPL/Software distributed under the License is distributed on an "ASIS" basis, WITHOUT WARRANTY OF ANY KIND, either express orimplied. See the License for the specific language governingrights and limitations under the License.The Original Code is the MPI Arbitrary Precision Integer Arithmeticlibrary.The Initial Developer of the Original Code is Michael J. Fromberger <sting@linguist.dartmouth.edu>Portions created by Michael J. Fromberger are Copyright (C) 1998, 2000 Michael J. Fromberger. All Rights Reserved.Contributor(s):Alternatively, the contents of this file may be used under theterms of the GNU General Public License Version 2 or later (the"GPL"), in which case the provisions of the GPL are applicableinstead of those above.  If you wish to allow use of yourversion of this file only under the terms of the GPL and not toallow others to use your version of this file under the MPL,indicate your decision by deleting the provisions above andreplace them with the notice and other provisions required bythe GPL.  If you do not delete the provisions above, a recipientmay use your version of this file under either the MPL or the GPL.$Id: expt.txt,v 1.1 2000/07/14 00:44:32 nelsonb%netscape.com Exp $