**Cryptography Tutorials - Herong's Tutorial Examples** - Version 5.32, by Dr. Herong Yang

RSA Public Key Encryption Algorithm

This section describes the RSA public key encryption algorithm. Generating public and private keys used in RSA encryption requires two large prime numbers.

RSA public key encryption algorithm was invented in 1976 by three MIT mathematicians, Ronald L. Rivest, Adi Shamir, and Leonard M. Adleman. The name of the algorithm "RSA" represents the initials of their surnames.

The first part of the RSA algorithm is the public key and private key generation, which can be described as:

- Choose two distinct prime numbers p and q. For security purposes, the integers p and q should be chosen at random, and should be of similar bit-length of 1024 bits or higher
- Compute n = p*q. n is used as the modulus for both public and private keys. Its length, usually expressed in bits, is the key length.
- Compute m = (p-1)*(q-1). m is actually the Euler's totient function value of n.
- Choose an integer e such that 1 < e < n and greatest common divisor gcd(e, m) = 1; i.e., e and m are coprime numbers.
- Compute d such that d*e mod m = 1. d is also called the modular multiplicative inverse of e with modulo m.
- Package the public key as {n,e}.
- Package the private key as {n,d}.

The second part of the RSA algorithm is the message encryption and decryption, which can be described as:

To encrypt a message, the sender can follow these steps:

- Divide the original message into blocks so that each block can be converted to a number, M < n.
- Compute the encrypted block with the public key as C = M**e mod n.
- Deliver encrypted blocks as the encrypted message to the owner of the private key.

To decrypt a message, the owner of the private key can follow these steps:

- Divide the encrypted message back into blocks of the same block size used in the encryption process.
- Decrypt the block with the private key as M = C**d mod n.
- Put decrypted block together to get the original message.

*Last update: 2013.*

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"

Using Certificates in IE (Internet Explorer)