expt.txt (3496B)
1 Exponentiation 2 3 For exponentiation, the MPI library uses a simple and fairly standard 4 square-and-multiply method. The algorithm is this: 5 6 Input: a, b 7 Output: a ** b 8 9 s = 1 10 11 while(b != 0) 12 if(b is odd) 13 s = s * a 14 endif 15 16 b = b / 2 17 18 x = x * x 19 endwhile 20 21 return s 22 23 The modular exponentiation is done the same way, except replacing: 24 25 s = s * a 26 27 with 28 s = (s * a) mod m 29 30 and replacing 31 32 x = x * x 33 34 with 35 36 x = (x * x) mod m 37 38 Here is a sample exponentiation using the MPI library, as compared to 39 the same problem solved by the Unix 'bc' program on my system: 40 41 Computation of 2,381,283 ** 235 42 43 'bc' says: 44 45 4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\ 46 4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\ 47 6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\ 48 4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\ 49 6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\ 50 FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\ 51 CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\ 52 5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\ 53 CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\ 54 49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\ 55 5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\ 56 A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\ 57 D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\ 58 92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\ 59 A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\ 60 AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\ 61 E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\ 62 1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\ 63 CFFF2E1AC93F3CA264A1B 64 65 MPI says: 66 67 4385CA4A804D199FBEAD95FAD0796FAD0D0B51FC9C16743C45568C789666985DB719\ 68 4D90E393522F74C9601262C0514145A49F3B53D00983F95FDFCEA3D0043ECEF6227E\ 69 6FB59C924C3EE74447B359B5BF12A555D46CB819809EF423F004B55C587D6F0E8A55\ 70 4988036A42ACEF9F71459F97CEF6E574BD7373657111648626B1FF8EE15F663B2C0E\ 71 6BBE5082D4CDE8E14F263635AE8F35DB2C280819517BE388B5573B84C5A19C871685\ 72 FD408A6471F9D6AFAF5129A7548EAE926B40874B340285F44765BF5468CE20A13267\ 73 CD88CE6BC786ACED36EC7EA50F67FF27622575319068A332C3C0CB23E26FB55E26F4\ 74 5F732753A52B8E2FB4D4F42D894242613CA912A25486C3DEC9C66E5DB6182F6C1761\ 75 CF8CD0D255BE64B93836B27D452AE38F950EB98B517D4CF50D48F0165EF0CCCE1F5C\ 76 49BF18219FDBA0EEDD1A7E8B187B70C2BAED5EC5C6821EF27FAFB1CFF70111C52235\ 77 5E948B93A015AA1AE152B110BB5658CB14D3E45A48BFE7F082C1182672A455A695CD\ 78 A1855E8781E625F25B41B516E77F589FA420C3B058861EA138CF7A2C58DB3C7504FD\ 79 D29554D78237834CC5AE710D403CC4F6973D5012B7E117A8976B14A0B5AFA889BD47\ 80 92C461F0F96116F00A97AE9E83DC5203680CAF9A18A062566C145650AB86BE4F907F\ 81 A9F7AB4A700B29E1E5BACCD6DCBFA513E10832815F710807EED2E279081FEC61D619\ 82 AB270BEB3D3A1787B35A9DD41A8766CF21F3B5C693B3BAB1C2FA14A4ED202BC35743\ 83 E5CBE2391624D4F8C9BFBBC78D69764E7C6C5B11BF005677BFAD17D9278FFC1F158F\ 84 1B3683FF7960FA0608103792C4163DC0AF3E06287BB8624F8FE3A0FFBDF82ACECA2F\ 85 CFFF2E1AC93F3CA264A1B 86 87 Diff says: 88 % diff bc.txt mp.txt 89 % 90 91 ------------------------------------------------------------------ 92 This Source Code Form is subject to the terms of the Mozilla Public 93 # License, v. 2.0. If a copy of the MPL was not distributed with this 94 # file, You can obtain one at http://mozilla.org/MPL/2.0/.