Cryptography Tutorials - Herong's Tutorial Examples - v5.42, by Herong Yang
Efficient RSA Encryption and Decryption Operations
This section describes an efficient way of carrying out RSA encryption and decryption operations provided by authors of RSA algorithm.
One efficient way to carry out the RSA encryption operation of "C = M**e mod n" is to use the following algorithm provided by authors of RSA as:
Step 1. Represent e in binary format and store it binary digits in array e[0], e[1], ..., e[k] Step 2. Set the variable C to 1 Step 3. For each i from 0 to k, repeat steps 3a and 3b Step 3a. C is reset to C*C mod n Step 3b. If e[i] = 1, C is reset to C*M mod n
The RSA decryption operation of "M = C**d mod n" can also be carried out using the same algorithm:
Step 1. Represent d in binary format and store it binary digits in array d[0], d[1], ..., d[k] Step 2. Set the variable M to 1 Step 3. For each i from 0 to k, repeat steps 3a and 3b Step 3a. M is reset to M**2 mod n Step 3b. If d[i] = 1, M is reset to M*C mod n
Let's use an example presented in the previous tutorial to validate the algorithm:
Given C = 62, d = 65, and n = 133 Calculate M = C**d mod n = 62**65 mod 133 Step 1. Represent d, 65, in binary format in array d[] d[] = {1,0,0,0,0,0,1}, k = 6 Step 2. Set the variable M to 1 M = 1 Step 3. For each i from 0 to 6, repeat steps 3a and 3b i = 0, d[0] = 1 M = 1*1 mod 133 = 1 (1) M = 1*62 mod 133 = 62 (2) i = 1, d[1] = 0 M = 62**2 mod 133 = 120 (3) i = 1, d[2] = 0 M = 120**2 mod 133 = 36 (4) i = 1, d[3] = 0 M = 36**2 mod 133 = 99 (5) i = 1, d[4] = 0 M = 99**2 mod 133 = 92 (6) i = 1, d[5] = 0 M = 92**2 mod 133 = 85 (7) i = 1, d[6] = 1 M = 85**2 mod 133 = 43 (8) M = 43*62 mod 133 = 6 (9)
Looks good. It matches the result presented in the previous tutorial.
If we start with the last calculation (9) and combine backward other calculations, we can see why this algorithm works:
M = 43*62 mod 133 # start with (9) = 85**2 *62 mod 133 # combine (8) = (92**2)**2 *62 mod 133 # combine (7) = ((99**2)**2)**2 *62 mod 133 # combine (6) = (((36**2)**2)**2)**2 *62 mod 133 # combine (5) = ((((120**2)**2)**2)**2)**2 *62 mod 133 # combine (4) = (((((62**2)**2)**2)**2)**2)**2 *62 mod 133 # combine (3), (2), (1) = 62**64 *62 mod 133 # consolidate exp. = 62**65 mod 133
Table of Contents
Introduction to AES (Advanced Encryption Standard)
DES Algorithm - Illustrated with Java Programs
DES Algorithm Java Implementation
DES Algorithm - Java Implementation in JDK JCE
DES Encryption Operation Modes
PHP Implementation of DES - mcrypt
Blowfish - 8-Byte Block Cipher
Secret Key Generation and Management
Cipher - Secret Key Encryption and Decryption
►Introduction of RSA Algorithm
What Is Public Key Encryption?
RSA Public Key Encryption Algorithm
Illustration of RSA Algorithm: p,q=5,7
Illustration of RSA Algorithm: p,q=7,19
Proof of RSA Public Key Encryption
►Efficient RSA Encryption and Decryption Operations
Proof of RSA Encryption Operation Algorithm
RSA Implementation using java.math.BigInteger Class
Introduction of DSA (Digital Signature Algorithm)
Java Default Implementation of DSA
Private key and Public Key Pair Generation
PKCS#8/X.509 Private/Public Encoding Standards
Cipher - Public Key Encryption and Decryption
OpenSSL Introduction and Installation
OpenSSL Generating and Managing RSA Keys
OpenSSL Generating and Signing CSR
OpenSSL Validating Certificate Path
"keytool" and "keystore" from JDK
"OpenSSL" Signing CSR Generated by "keytool"
Migrating Keys from "keystore" to "OpenSSL" Key Files
Certificate X.509 Standard and DER/PEM Formats
Migrating Keys from "OpenSSL" Key Files to "keystore"