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133 lines
4.9 KiB
Plaintext
133 lines
4.9 KiB
Plaintext
Exponentiation
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For exponentiation, the MPI library uses a simple and fairly standard
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square-and-multiply method. The algorithm is this:
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Input: a, b
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Output: a ** b
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s = 1
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while(b != 0)
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if(b is odd)
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s = s * a
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endif
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b = b / 2
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x = x * x
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endwhile
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return s
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The modular exponentiation is done the same way, except replacing:
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s = s * a
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with
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s = (s * a) mod m
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and replacing
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x = x * x
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with
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x = (x * x) mod m
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Here is a sample exponentiation using the MPI library, as compared to
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the same problem solved by the Unix 'bc' program on my system:
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Computation of 2,381,283 ** 235
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'bc' says:
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4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\
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4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\
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6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\
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4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\
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6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\
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FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\
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CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\
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5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\
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CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\
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49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\
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5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\
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A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\
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D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\
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92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\
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A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\
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AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\
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E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\
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1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\
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CFFF2E1AC93F3CA264A1B
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MPI says:
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4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\
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4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\
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6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\
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4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\
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6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\
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FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\
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CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\
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5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\
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CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\
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49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\
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5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\
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A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\
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D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\
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92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\
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A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\
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AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\
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E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\
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1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\
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CFFF2E1AC93F3CA264A1B
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Diff says:
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% diff bc.txt mp.txt
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%
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------------------------------------------------------------------
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***** BEGIN LICENSE BLOCK *****
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Version: MPL 1.1/GPL 2.0/LGPL 2.1
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The contents of this file are subject to the Mozilla Public License Version
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1.1 (the "License"); you may not use this file except in compliance with
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the License. You may obtain a copy of the License at
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http://www.mozilla.org/MPL/
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Software distributed under the License is distributed on an "AS IS" basis,
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WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
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for the specific language governing rights and limitations under the
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License.
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The Original Code is the MPI Arbitrary Precision Integer Arithmetic
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library.
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The Initial Developer of the Original Code is
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Michael J. Fromberger <sting@linguist.dartmouth.edu>
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Portions created by the Initial Developer are Copyright (C) 1998, 2000
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the Initial Developer. All Rights Reserved.
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Contributor(s):
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Alternatively, the contents of this file may be used under the terms of
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either the GNU General Public License Version 2 or later (the "GPL"), or
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the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
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in which case the provisions of the GPL or the LGPL are applicable instead
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of those above. If you wish to allow use of your version of this file only
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under the terms of either the GPL or the LGPL, and not to allow others to
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use your version of this file under the terms of the MPL, indicate your
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decision by deleting the provisions above and replace them with the notice
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and other provisions required by the GPL or the LGPL. If you do not delete
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the provisions above, a recipient may use your version of this file under
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the terms of any one of the MPL, the GPL or the LGPL.
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***** END LICENSE BLOCK *****
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$Id: expt.txt,v 1.2 2005/02/02 22:28:22 gerv%gerv.net Exp $
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