Asymmetric encryption and decryption-RSA Algorithm

Generation of public and private keys

N = 17 * 53 = 901m = (17-1)*(53-1)=832let e=17 // by random d = 49 // because ( d * e ) % m = 1 , that is ( d * 17 ) % 832 = 1

Public Key (901,17) private key (901,49)

Encryption Process

Int text = 123 (note that the message must be smaller than N );

Because text ^ e % N = code, that is, 123 ^ 17% 901 = Code

Get code = 378

Decryption process

Int code = 378;

Because code ^ d % N = text, that is, 378 ^ 49% 901 = text

Get text = 123

Continue test text = 456, text = 789

pow=337587917446653715596592958817679803remain is encrypted msg, 378pow=1981975090781375567648692626283639181924626345824851461901652399140883944715714801363950285365649593405611421793924739086942208remain is decrypted msg, 123pow=1593684413895892775431288016392119629314523136remain is encrypted msg, 31pow=11932267966476130911517907186867185074148138371734008968629799047943267871remain is decrypted msg, 456pow=17795453274736834362412998969384875211516379055829remain is encrypted msg, 24pow=42692426938810341355395254247148178158685421582623594902764046516224remain is decrypted msg, 789

If the random private key is 23, obtained from (D * E) % 832 = 1, D = 1447

Key pair (901,23) (901,1447)

Encryption process 2

Int text = 123 (note that the message must be smaller than N );

Because text ^ e % N = code, that is, 123 ^ 23% 901 = Code

Get code = 149

./rsa 123 23 901

pow=1169008215014432917465348578887506800769541157267remain=149

Decryption process 2

Int code = 149;

Because code ^ d % N = text, that is, 149 ^ 1447% 901 = text

Get text = 123

./rsa 149 1447 901pow=3985936655027786133408829502155172486035904622712472852621608241746913310297391658184716646894437322554704092666282273452855275384085318288046586224429347497827880247547339473036917816909223107117175531587260639648846353409844711958706092093596730245565996912003513233565007065727967455584025260911486810746630678094905762197449698418393359848248040053204101912486274919097714199485366154899138632699959288422245852748033196204793416492537560971606021510788711927751466478702987029084746015114294265995598688131965702319407549561404014161346291768678253671588393974329706005805322594725598901652699451931023169921356303019131994689149252324743527490479491940886193543681983109194838286138082417166699477848202718347964687820796592019328071391913496121999081649165198851436538061394347514783202440077109779016851948752385464899137550781743050874740569187871261877899262047556241037569513727241777013958245104990221096084832705392187584307190592892229359969096765098092852369655657703699448285550244408917771734036376812464079731831055765945558390839789584335008109083209997253493029025530744552730080488782762304788784814726079959495953838029404233253982382231821825059796386932843561833899624385090559266388309444170895176942274476213381834239263776359183712755409616899578532221617869355630180167696863598518217920559059880617368830175766115161601439972079153451714153741788486717251510899260246046823014371599692024285779413561563126708199148100439542764252953767607725285544447473028149388111275729639654549626974834654173824045613468737321214142959704129737498708807684383933171830693895907145483744154610946805717856953750819840697171152907340638764324070540105234490652588160359246349466614681404427878668385500329188714136429101735281721801482418741695141767516243529016466547669220693256890471638180522587977329481861127690785609948263132745032373176370846812221619669962814451607786769451764283956667048224778080078119822225893910169294528225164973914039173067841802331919352006767688214788854741209476129408348246067157963999292702077540465878741199438072301716437512437206906827652022918225801876338146245435414504850134452701815858979786932044372642085983768077238865428181442964225497161921056275373467702146501770271202169524135998423744124666111748049368433360408756284224496271587223056244362621755475005116811157719670973646470109741830322978187377451183707774175105440143172725174869125803081026121461473916991813456601031329814195352576206287933278596520860904193983118052847549233481720087449879783102557667902374897959615060335464554488838194211827049636607371477869631641345731986506361643642064555937668953135519546086134667128658277195852003986966590034063953363008297418066438616213070509869485074973762746760573414761610051773364844423773838759134377078566230721304823672091660155346094353132676209154505265788271907294843328075680525047092758196227733579042369746326640103471517003588082621099265771434203691296983507676617026988511984284930947929603040015756972741093414371038843713030209328408570065664368532865353731976350820941739288708181549678167180213092260955630329359228466794631319443883593072931779043019549remain=123

Optimization of decryption (Chinese Remainder Theorem)

Dp = D Mod (p-1) = 1447 Mod (17-1) = 7dq = D Mod (Q-1) = 1447 Mod (53-1) = 43qinv * q mod p = 1 that is, qinv * 53 mod 17 = 1 get qinv = 9

m1=code^dp mod p = 149^7   mod17  = 4m2=code^dq mod q = 149^43 mod 53 = 17h= (qInv * (m1-m2))mod p = (9 * (-13)) mod 17 = 2r= m2 + h*q = 17 + 2*53= 17 + 106 = 123 

The result is the same as that obtained by the above optimization solution.

Demo program

It is written in C language. The big data computing library uses an Open Source GMP library, as shown below:

Http://gmplib.org/

The most common algorithm is the M modulo code of the N power of X.

#include <stdio.h>#include <stdlib.h>#include <gmp.h>int main(int argc, char * argv[]){mpz_t pow, mod;mpz_init(pow);mpz_init(mod);mpz_ui_pow_ui(pow, atoi(argv[1]), atoi(argv[2]));printf("pow=");mpz_out_str(stdout, 10, pow);printf("\n");mpz_mod_ui(mod, pow, atoi(argv[3]));printf("remain=");mpz_out_str(stdout, 10, mod);printf("\n");return 0;}

Project File

Http://hi.csdn.net/attachment/201201/20/0_1327043412h7wW.gif