RSA Encryption & Cryptography

Ryan KoonMariko Nihonyanagi

Lynette Rota

Definition Cryptography is the use of math for

information security. This includes data integrity, confidentiality,

entity authentication, and data origin authentication.

Today’s cryptography depends on the existence of difficult algorithmic problems.

The word cryptology comes from the Greek word kryptós logos, meaning ‘hidden word’.

Biographies

Three inventers of RSA Public-key Crytposystem. Ronald L. Rivest – PHD from Stanford, In 2007 he

received the Computers, Freedom and Privacy Conference "Distinguished Innovator" award.

Adi Shamir – PHD from the Weizmann Institute, recipient of many awards including the IEEE's W.R.G. Baker Prize, holder of many patents and publications.

Len Adleman – PHD from UC Berkley, known for first use of DNA to compute an algorithm.

Public Key Encryption

RSA Background RSA is one of the best-known public-key

cryptosystems. Cryptography itself has a long history, dating

from some 4,000 years ago in ancient Egypt. In 1977, the U.S. adopted DES, the Data

Encryption Standard for use with all unclassified information.

RSA now uses the digital signature, which was first standardized in 1991.

Research in cryptography continues to this day.

Real World Cryptography Applications RSA can be used for secure communication (e.g. cellular

phones), identification, authentication (e.g. digital signatures), secret sharing, electronic commerce, certification, secure electronic mail, key recovery, and secure computer access.

Other uses of cryptography include e-mail, online banking, online trading, online credit cards transactions, satellite and cable television.

RSA can be used at automatic teller machine (ATM) machines. The ATM card, or the public key is associated with the personal identification number (PIN), or the private key.

It is used for Ethernet network cards, smart cards, secure telephones, and protocols for secure Internet communications

RSA is a standard cryptosystem that has been used by over 700 companies.

RSA Devices RSA is used for

Secure ID’s, only allowing access for a limited amount of time.

The RSA CipherGenerate your Private and Public Keys1. RSA Cryptography is an asymmetric encryption system (also known as

public-key cryptography) meaning that the formulas used to derive the encrypted and decrypted messages use different keys. Below is RSA’s way of generating the private and public keys for the decryption and encryption algorithms.

2. Choose two different, very large prime numbers p, and q.3. Computer n = p*q

n is used as the modulus for the public and private keys.4. Compute the totient using n, p and q: Ø(n) = (p-1)(q-1).5. Choose an integer e such that 1<e< Ø(n) , and e and Ø(n) share no

factors other than 1 (i.e. e and φ(n) are coprime) 1. e is released as the public key exponent

6. Compute d to satisfy the congruence relation de ≡ 1 (mod Ø(n)) ; i.e. for some integer de = 1+kØ(n) .

d is kept as the private key exponentThe public key consists of the modulus n and the public (or encryption)

exponent e.The private key consists of the modulus n and the private (or decryption)

exponent d which must be kept secret.

The RSA Cipher

Encryption/DecryptionEncryption Step 1: Transmit your public key (n & e) Step 2: Sender changes their message into a number such

that the resulting number is less than p*q. This is done through a hash function, using a hash table

Step 3: Sender generates their cipher-text using the RSA encryption formula: c=me mod n

Step 4: Transmit result to recipient.

Decryption (Assuming a transmitted message using your public key has

been received) Perform: m=cd mod n In this case, d is the private key that the recipient

withheld. The result will be the original number the sender had before they input it into the encryption formula.

Security of RSA RSA is an extremely secure cipher system because the

numbers chosen to determine its asymmetric keys are extremely large prime numbers.

Some can reach up to 1000 decimal digits, or over 3000 bits in binary.

Decrypting an RSA cipher-text is thought to be infeasible at best. Partial decryptions can be possible, but well made padding schemes protect cipher-texts in this way.

Additionally, attempting to find out what numbers were chosen originally would let you solve the decryption near instantly.

However, attempting to find out which numbers create a certain modulus is very difficult since in modular arithmetic many numbers (coprime or not) have a congruence relationship.

Currently, RSA offers prize money to individuals who can find the factors to crack their private key ciphers.

Demonstration http://

www.securecottage.com/demo/rsa2.html

http://www.gax.nl/wiskundePO/#

References (2007). RSA laboratories. Retrieved November 12, 2007, from http://www.rsa.com/rsalabs/ Daubechies, I.. (2003). Math alive. Princeton University. Retrieved November 12, 2007, from

http://www.math.princeton.edu/matalive/Crypto/ Menezes, A. J., van Oorschot, P. C., & Vanstone, S. A. (2001). Handbook of applied

cryptography. University of Waterloo. Retrieved November 12, 2007, from http://www.cacr.math.uwaterloo.ca/hac/

Ray, I. (2006). Public key cryptography. Colorado State University. Retrieved November 12, 2007, from http://www.cs.colostate.edu/~cs556dl/lecture-notes/rsa.pdf

Rivest, R. L. (2007). Ronald L. Rivest: HomePage. massachusetts institute of technology. Retrieved November 12, 2007, from http://people.csail.mit.edu/rivest/

Wikipedia Users. (2007). Ron Rivest. Wikipedia: The Free Encyclopedia. Retrieved November 12, 2007, from http://en.wikipedia.org/wiki/Ron_Rivest

Wikipedia Users. (2007). RSA. Wikipedia: The Free Encyclopedia. Retrieved November 12, 2007, from http://en.wikipedia.org/wiki/RSA

Images: (2007). Computer ethernet cable connection. iStock International Inc. Retrieved November 12,

2007, from http://www.istockphoto.com/file_closeup/?id=3597040&refnum=1816354 Bowen, J. (2007). How ATMs work. HowStuffWorks, Inc. Retrieved November 12, 2007, from

http://money.howstuffworks.com/atm1.htm Cherowitzo, B. (2001). The University of Colorado Denver. Retrieved November 12, 2007, from

http://www-math.cudenver.edu/~wcherowi/clock.gif Higgs, B. J. (2006). Cryptography through the ages: A layman's view. University of Basel.

Retrieved November 12, 2007, from http://informatik.unibas.ch/lehre/ss07/cs221/_Downloads/cs221-20070615-history-6up.pdf