Presented byKatherine Heller

COSC 4765University of WyomingApril 26, 2011

RSA Asymmetric Key Cryptosystem

Image source: PC Dynamics, Inc.

Introduced 1970’s

Whitfield Diffie and Martin Hellman

Known as Public Key Encryption (PKE)

Eliminated need for shared private keys

Asymmetric Key Cryptography

Rivest, Shamir and Adleman

First asymmetric encryption algorithm

Encryption and authentication

Used with DES, SSL, CDPD and PGP

Most widely used asymmetric cipher

RSA

A function (F)+

A plaintext message (m)+

An encryption key (k)=

Ciphertext (c)

Encryption

Two keys: one public (kp)one private (ks)

F(m, kp) = c and F-1(c, ks) = m

F-1(F(m, kp), ks) = m

The RSA Method

Select two large prime numbers: p and q. Find the product, n, of p and q: n = pq. Choose a number, e, which is less than n and

relatively prime to (p-1)(q-1). Find a number d, such that (ed - 1) is evenly

divisible by (p-1)(q-1). e is the public exponent, d is the private

exponent. Public key: (n, e) Private key: (n, d)

The RSA Algorithm

Using real numbers:

p = 5077 and q = 4999

n = pq = 25379923e = 5( p – 1 ) = ( 5077 – 1 ) = 5076( q – 1 ) = ( 4999 – 1 ) = 49985076 * 4998 = 25369848d = 15221909( 5 (15221909) – 1 ) / 25369848 = 3

The RSA Algorithm (2)

What are the keys?

n = 25379923, e = 5 and d = 15221909

Public Key is the pair (n, e) or (25379923, 5)Used to encrypt

Private Key is the pair (n, d) or (25379923, 15221909)

Used to decrypt

Keys

Creating the ciphertextc = me mod n

Decrypting the messagem = cd mod n

Remember, n is really, really huge!

Keys (2)

Larger modulus (n) increases security Large keys Commonly 1024, 2048 and 4096 bits Keys ≥ 2048 bits for extremely

valuable data Difficult to compare to other methods Security comes from how the keys are

generated, as well as key length

Key Sizes

Produces ciphertext without patterns

Very random

Hard to exploit

Larger modulus = greater security

What’s so good about RSA?

Modular exponentiation slows it down

Longer key = slower operations

◦ 2 x modulus ⇒ time for public key ops x 4time for private key ops x 8

time for key generation x 16

◦ Public key ops take O(k2) steps◦ Private key ops take O(k4) steps (where k = number of bits in modulus n)

DES 1000 times faster

But, how fast is it?

The de facto standard for cryptography

Combines authentication with encryption

Allows world-wide use of one system regardless of software or platforms

The Standard

Digital Envelope

LARGE PRIME NUMBERS

100 digits long, or longer (each!)

Factoring very difficult

Security in the mathematical difficulty

Resistant to key search attacks

The “Key” to Security

RSA can still be broken, with the key

Discovering a private key corresponding to its paired public key

“Guessed Plaintext Attack”◦ Guess the message◦ Run the encryption to see if it matches ciphertext

Even so – RSA isn’t going anywhere

And with the key…

RSA Algorithm Demo by Richard Holowczak:http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/#overview

RSA.com FAQ document:http://www.rsa.com/rsalabs/node.asp?id=2152#

More information:

Coated.com. (2010). GSM Security Encryption Code Hacked. Retrieved April 23, 2011, from Coated.com: http://www.coated.com/gsm-security-encryption-code-hacked-93620004/

Daswani, N., Kern, C., & Kesavan, A. (2007). Foundations of Security: What Every Programmer Needs to Know. Berkeley: Apress.

PC Dynamics, Inc. (2011). File Encryption. Retrieved April 23, 2011, from SafeHouseSoftware.com: http://www.safehousesoftware.com/FileEncryption.aspx

Richard Holowczak, P. (2002, September 12). RSA Demo Applet. Retrieved April 16, 2011, from cisnet.baruch.cuny.edu: http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/#overview

RSA Laboratories. (2000). RSA Laboratories' Frequently Asked Questions About Today's Cryptography, Version 4.1. Retrieved April 16, 2011, from RSA Laboratories: http://www.rsa.com/rsalabs/node.asp?id=2152#

Welschenbach, M. (2005). Cryptography in C and C++. New York: Apress.

References

Questions?

Image source: Coated.com