Cryptography Tutorials - Herong's Tutorial Examples - v5.42, by Herong Yang
Proof of RSA Encryption Operation Algorithm
This section discusses how to prove the RSA encryption operation algorithm.
We can also prove the encryption operation algorithm presented in the previous section in the following way:
Given integers M, e, and n Express e as a binary expression: e = e[0]*2**k + e[1]*2**(k-1) + ... + e[k-1]*2**1 + e[k]*2**0 Given C = M**e mod n Replace e with the binary expression: C = M**(e[0]*2**k + e[1]*2**(k-1) + ... + e[k-1]*2**1 + M**e[k]*2**0) mod n Split and simply the e[k] term: C = M**(e[0]*2**k + e[1]*2**(k-1) + ... ) * M**e[k] mod n Factor remaining terms: C = (M**(e[0]*2**(k-1) + e[1]*2**(k-2) + ... + e[k-1]*2**0))**2 * M**e[k] mod n Split and simply the e[k-1] term: C = (M**(e[0]*2**(k-1) + e[1]*2**(k-2) + ...) * M**e[k-1])**2 * M**e[k] mod n Repeat last two steps for e[k-2], ... e[0] terms: C = ((...((M**e[0])**2 * M**e[1])**2 * ...)**2 * M**e[k-1])**2 * M**e[k] mod n Move "mod n" into each team: C = ((...((M**e[0] mod n)**2 * M**e[1] mod n)**2 * ...)**2 * M**e[k-1] mod n)**2 * M**e[k] mod n
As you can see from the last expression, the "(...)**2 * M**e[i] mod n" is a repeating pattern. This leads us to the following algorithm to calculate "C = M**e mod n":
Given integers M, e, and n Calculcate "C = M**e mod n": 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 step 3a Step 3a. C is reset to "C**2 * M**e[i] mod n"
This algorithm is identical to the algorithm presented in the previous section, because "* M**e[i]" is optional if e[i]=0.
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"