CRYPTOGRAPHY

RSA Public Key Algorithm

RSA Algorithm history

Invented in 1977 at MIT

Named for Ron Rivest, Adi Shamir, and Len Adleman

Based on 2 keys, 1 public and 1 private

Basic Algorithm

A private and a public key are generated by a user.

The public key is given to a sender, who encrypts the message using that key.

The user then decrypts the message using the private key

Key generation 2 large prime numbers P and Q are randomly

chosen N = P * Q S = (P-1) (Q-1) Value E is chosen where 1<E<N and the

greatest common denominator of E and S is 1 A value D is calculated where (D*E) modS = 1 D is the private key (N,E) is published as the public key

Encryption

To send a message a person uses the public key

The plaintext message is broken into binary segments no larger than N

Each segment is encrypted with the algorithm ciphertext = plaintextE mod N

Decryption

To decrypt the message, the recipient will calculate each segment with plaintext = ciphertextD mod N

Confused?

Alfred chooses 2 primes

P = 19 and Q = 31

N = P*Q = 589

Alfred finds e

(P-1)(Q-1) = 540

E needs to have no GCD with 540

Can be found with Euclid’s algorithm (Example 27.6, p614 Automata)

E = 49

Alfred finds D

D is computed using an extension of Euclid’s algorithm

D = 1069

1069 is the private key

(589,49) is the public key

Batman wants to send message

The message will be “A” ASCII code for A is 65 The encryption algorithm is then:

6549 mod 589 However, don’t actually need to compute

6549

Batman exploits 2 facts

Ni+j = Ni * Nj

(N*M) mod K = (N mod K)(M mod K) mod K

This is called modular exponentiation

Note that 6549 = 651+16+32

Batman encrypts message and sends

6549 mod 589 = 651+16+32 mod 589 (651*6516*6532) mod 589 (651 mod 589)(6516 mod 589)(6532 mod

589) mod 589 (65 * 524 * 102) mod 589 3474120 mod 589 = 198 Batman sends Alfred the message 198

Alfred decrypts message

Alfred uses the private key 1069 and computes 1981069 mod 589

This is done with the same process used in encryption to retrieve the message “A”

Why it’s effective

Larger integers increase effectiveness Encryption and decryption are inverses

of each other User A can find E and D efficiently Modular exponentiation allows both

users to compute encryption and decryption efficiently

Why it’s effective

Any eavesdropper will not be able to recreate the enciphered text because the modular exponentiation is not an invertible function

Also, the private key D cannot be calculated from the public keys N and E

References

Automata, Computability, and Complexity. Elaine Rich. Pearson Prentice Hall, 2008.

RSA Laboratories' Frequently Asked Questions About Today's Cryptography, Version 4.1. RSA Laboratories Inc, 2000. http://www.rsa.com/rsalabs/node.asp?id=2152#