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Exponentiation
For exponentiation, the MPI library uses a simple and fairly standard
square-and-multiply method. The algorithm is this:
Input: a, b
Output: a ** b
s = 1
while(b != 0)
if(b is odd)
s = s * a
endif
b = b / 2
x = x * x
endwhile
return s
The modular exponentiation is done the same way, except replacing:
s = s * a
with
s = (s * a) mod m
and replacing
x = x * x
with
x = (x * x) mod m
Here is a sample exponentiation using the MPI library, as compared to
the same problem solved by the Unix 'bc' program on my system:
Computation of 2,381,283 ** 235
'bc' says:
4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\
4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\
6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\
4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\
6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\
FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\
CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\
5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\
CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\
49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\
5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\
A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\
D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\
92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\
A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\
AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\
E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\
1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\
CFFF2E1AC93F3CA264A1B
MPI says:
4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\
4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\
6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\
4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\
6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\
FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\
CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\
5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\
CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\
49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\
5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\
A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\
D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\
92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\
A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\
AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\
E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\
1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\
CFFF2E1AC93F3CA264A1B
Diff says:
% diff bc.txt mp.txt
%
------------------------------------------------------------------
The contents of this file are subject to the Mozilla Public
License Version 1.1 (the "License"); you may not use this file
except in compliance with the License. You may obtain a copy of
the License at http://www.mozilla.org/MPL/
Software distributed under the License is distributed on an "AS
IS" basis, WITHOUT WARRANTY OF ANY KIND, either express or
implied. See the License for the specific language governing
rights and limitations under the License.
The Original Code is the MPI Arbitrary Precision Integer Arithmetic
library.
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 the
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"GPL"), in which case the provisions of the GPL are applicable
instead of those above. If you wish to allow use of your
version of this file only under the terms of the GPL and not to
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indicate your decision by deleting the provisions above and
replace them with the notice and other provisions required by
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may use your version of this file under either the MPL or the GPL.
$Id$
|
