Illustration of RSA Algorithm: p,q=5,7

This section provides a tutorial example to illustrate how RSA public key encryption algorithm works with 2 small prime numbers 5 and 7.

To demonstrate the RSA public key encryption algorithm, let's start it with 2 smaller prime numbers 5 and 7.

Generation the public key and private key with prime numbers of 5 and 7 can be illustrated as:

Given p as 5
Given q as 7
Compute n = p*q: n = 5*7 = 35
Compute m = (p-1)*(q-1): m = 4*6 = 24
Select e, such that e and m are coprime numbers: e = 5
Compute d, such that d*e mod m = 1: d = 29
The public key {n,e} is = {35,5}
The private key {n,d} is = {35,29}

With the public key of {35,5}, encryption of a cleartext M represented as number 23 can be illustrated as:

Given public key {n,e} as {35,5}
Given cleartext M represented in number as 23
Divide B into blocks: 1 block is enough
Compute encrypted block C = M**e mod n: 
   C = 23**5 mod 35 = 6436343 mod 35 = 18
The ciphertext C represented in number is 18

With the private key of {35,29}, decryption of the ciphertext C represented as number 18 can be illustrated as:

Given private key {n,e} as {35,29}
Given ciphertext C represented in number as 18
Divide C into blocks: 1 block is enough
Compute encrypted block M = C**d mod n: 
   M = 18**29 mod 35 
     = 18*18**28 mod 35
     = 18*(18**4)**7 mod 35
     = 18*(104976)**7 mode 35
     = 18*(104976 mod 35)**7 mod 35
     = 18*(11)**7 mod 35
     = 18*19487171 mod 35
     = 350769078 mod 35
     = 23
The cleartext M represented in number is 23

Cool. RSA public key encryption algorithm works. We are getting the original cleartext 23 back using the decryption algorithm and private key!

Notice that I had to compute "18**29 mod 35" in multiple steps, because 18**29 is too big to be computed directly.

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