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"