RSA Public Key Crypto System

1. RSA Public Key Crypto System

2. About RSA • Announced in 1977 by Ronald Rivest, Adi Shamir, and Leonard Adleman • Relies on the relative ease of finding large primes and the relative difficulty of factoring integers.

3. RSA Public Key and Encryption • n = pq, two large primes p and q are kept secret. • Choose a random integer e which is relatively prime to (p-1)( q-1). • Public key is the pair (n,e) • Encryption E: C = E(B) = B e (mod n)

4. RSA Private Key and Decryption • Calculate an integer d such that ed is congruent to 1 (mod (p-1)( q-1)). • The pair (n,d) is the private key. • Decryption D: D(C) = C d(mod n).

5. Breaking RSA • No known way to break the RSA system without finding the prime factorization of n. • As factorization methods continue to improve and computer power continues to increase, the key sizes used in RSA encryption must also be increased. • In 1977 Rivest, Shamir and Adleman published a challenge (a message encrypted) using 129 digit integers. They expected this to remain unbroken for a long time. But it was broken in 1994 using about the same amount of computer operations used to animate the movie Toy Story.

6. RSA Example • Pick the primes p=3457631 and q=4563413. • Choose the exponent e=1231239 • Public Key = (n, e) = (15778598254603, 1231239). • To encrypt the message “George has green hair” we convert it to an integer. Using, for example, the encoding a=01, b=02, . . . , z = 26, blank=27 we get 0705151807052 7080119270718 0505142718010 918. • For each of the four blocks (whose length was chosen so the blocks would represent integers no larger than n) we compute Be (mod n) (using the binary exponentiation). This gives the encrypted message: 1658228449402 5333403068473 7979527536648 13889903320423. • This message can be decrypted by raising each block to the 1315443185039th power modulo n.