**Cryptography Tutorials - Herong's Tutorial Examples** - v5.42, by Herong Yang

Illustration of RSA Algorithm: p,q=7,19

This section provides a tutorial example to illustrate how RSA public key encryption algorithm works with 2 small prime numbers 7 and 19.

As the second example to illustrate how RSA public key encryption algorithm works, let's take prime numbers of 7 and 19 as presented by Paul Johnston at http://pajhome.org.uk/crypt/rsa/rsa.html.

Generation the public key and private key with prime numbers of 7 and 19 can be illustrated as:

Given p as 7 Given q as 19 Compute n = p*q: n = 7*19 = 133 Compute m = (p-1)*(q-1): m = 6*18 = 108 Select e, such that e and m are coprime numbers: e = 5 Compute d, such that d*e mod m = 1: d = 65 The public key {n,e} is = {133,5} The private key {n,d} is = {133,65}

With the public key of {133,5}, encryption of a cleartext M represented as number 6 can be illustrated as:

Given public key {n,e} as {133,5} Given cleartext M represented in number as 6 Divide M into blocks: 1 block is enough Compute encrypted block C = M**e mod n: C = 6**5 mod 133 = 7776 mod 133 = 62 The ciphertext c represented in number is 62

With the private key of {133,65}, decryption of the ciphertext represented as number 62 can be illustrated as:

Given private key {n,e} as {133,65} Given ciphertext C represented in number as 62 Divide C into blocks: 1 block is enough Compute encrypted block M = C**d mod n: M = 62**65 mod 133 = 62*62**64 mod 133 = 62*(62**4)**16 mod 133 = 62*(14776336)**16 mode 133 = 62*(14776336 mod 133)**16 mod 133 = 62*(36)**16 mod 133 = 62*(36**4 mod 133)**4 mod 133 = 62*(92)**4 mod 133 = 4441636352 mod 133 = 6 The cleartext M represented in number is 6

Very nice. We are getting the original cleartext 6 back with no problem with the private key.

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"