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

Last update: 2013.

Table of Contents

 About This Book

 Cryptography Terminology

 Cryptography Basic Concepts

 Introduction to AES (Advanced Encryption Standard)

 Introduction to DES Algorithm

 DES Algorithm - Illustrated with Java Programs

 DES Algorithm Java Implementation

 DES Algorithm - Java Implementation in JDK JCE

 DES Encryption Operation Modes

 DES in Stream Cipher 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

 How Secure Is RSA Algorithm?

 How to Calculate "M**e mod n"

Efficient RSA Encryption and Decryption Operations

 Proof of RSA Encryption Operation Algorithm

 Finding Large Prime Numbers

 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

 MD5 Mesasge Digest Algorithm

 SHA1 Mesasge Digest Algorithm

 OpenSSL Introduction and Installation

 OpenSSL Generating and Managing RSA Keys

 OpenSSL Managing Certificates

 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"

 Using Certificates in IE (Internet Explorer)

 Using Certificates in Firefox

 Using Certificates in Google Chrome

 Outdated Tutorials

 References

 PDF Printing Version