michael@0: Exponentiation
michael@0: 
michael@0: For exponentiation, the MPI library uses a simple and fairly standard
michael@0: square-and-multiply method.  The algorithm is this:
michael@0: 
michael@0: Input:	a, b
michael@0: Output: a ** b
michael@0: 
michael@0: 	s = 1
michael@0: 
michael@0: 	while(b != 0)
michael@0: 	  if(b is odd)
michael@0: 	    s = s * a
michael@0: 	  endif
michael@0: 
michael@0: 	  b = b / 2
michael@0: 
michael@0: 	  x = x * x
michael@0: 	endwhile
michael@0: 
michael@0: 	return s
michael@0: 
michael@0: The modular exponentiation is done the same way, except replacing:
michael@0: 
michael@0: 	s = s * a
michael@0: 
michael@0: with
michael@0: 	s = (s * a) mod m
michael@0: 
michael@0: and replacing
michael@0: 
michael@0: 	x = x * x
michael@0: 
michael@0: with
michael@0: 
michael@0: 	x = (x * x) mod m
michael@0: 
michael@0: Here is a sample exponentiation using the MPI library, as compared to
michael@0: the same problem solved by the Unix 'bc' program on my system:
michael@0: 
michael@0: Computation of 2,381,283 ** 235
michael@0: 
michael@0: 'bc' says:
michael@0: 
michael@0: 4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\
michael@0: 4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\
michael@0: 6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\
michael@0: 4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\
michael@0: 6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\
michael@0: FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\
michael@0: CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\
michael@0: 5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\
michael@0: CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\
michael@0: 49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\
michael@0: 5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\
michael@0: A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\
michael@0: D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\
michael@0: 92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\
michael@0: A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\
michael@0: AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\
michael@0: E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\
michael@0: 1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\
michael@0: CFFF2E1AC93F3CA264A1B
michael@0: 
michael@0: MPI says:
michael@0: 
michael@0: 4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\
michael@0: 4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\
michael@0: 6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\
michael@0: 4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\
michael@0: 6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\
michael@0: FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\
michael@0: CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\
michael@0: 5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\
michael@0: CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\
michael@0: 49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\
michael@0: 5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\
michael@0: A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\
michael@0: D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\
michael@0: 92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\
michael@0: A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\
michael@0: AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\
michael@0: E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\
michael@0: 1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\
michael@0: CFFF2E1AC93F3CA264A1B
michael@0: 
michael@0: Diff says:
michael@0: % diff bc.txt mp.txt
michael@0: %
michael@0: 
michael@0: ------------------------------------------------------------------
michael@0:  This Source Code Form is subject to the terms of the Mozilla Public
michael@0:  # License, v. 2.0. If a copy of the MPL was not distributed with this
michael@0:  # file, You can obtain one at http://mozilla.org/MPL/2.0/.