1. IS 302: Information Security and Trust Week 4: Asymmetric Encryption
2012

2. Review Symmetric block ciphers DES 3-DES AES ECB, CBC
© Yingjiu Li 2007

3. key distribution and key management
© Yingjiu Li 2007

4. Asymmetric Cryptosystem
private key d public key e
Encryption: C=E(e, P)
Decryption: P=D(d, C)
Alice e
Bob d
private
public
Plaintext
Encryption Algorithm E
Ciphertext
Decryption Algorithm D
Plaintext P C P
Computationally infeasible
© Yingjiu Li 2007

5. Asymmetric Encryption
Plaintext
Public key e
Encrypt
Private key d
Decrypt
Plaintext
© Yingjiu Li 2007

6. RSA RSA cryptosystem Rivest-Shamir-Adelman in 1978
Turing award in 2002
© Yingjiu Li 2007

7. RSA Keys Public keys (n,e), private keys d
n: a composite. n=pq, where p and q are large primes
d: an integer, 2<d<n
e: an integer, 2<e<n, s.t. for any number x, x=(xe mod n) d
mod n
© Yingjiu Li 2007

8. RSA Encryption/Decryption
Encryption of P: C=Pe mod n
Decryption of C: P=Cd mod n
PKCS: P =plaintext+ random padding < n
Why need random padding?
C=Pe mod n d
n,e
n,e
Bob
insecure channel
Alice
d: Bob’s private key
n,e: Bob’s public key
© Yingjiu Li 2007

9. How to Choose Keys in RSA
Choose 2 large prime numbers p, q
Compute n=pq and φ=(p-1)(q-1)
Choose e relatively prime to φ
Compute d from φ and e such that e*d mod φ =1
Public: n, e
Private: p, q, φ, d
A Toy Example:
Let p = 47, q = 71, then n = pq = 3337, φ=(p-1)(q-1) = 3220
Let e =79, Note φ = 3220 = 22  5  7  23
Solve d from d*79 mod 3220 =1, which gives d = 1019 (using extended Euclidean algorithm)
Let P = 688, then C =Pe mod n = 688^79 mod 3337 = 1570
P = Cd mod n = 1570^1019 mod 3337
© Yingjiu Li 2007

10. RSA Demo Cryptool  indiv. Procedures  RSA
Generate prime numbers
Factorization of a number
Cryptool  Encrypt/decrypt 
RSA encryption (in blocks)
RSA decryption
© Yingjiu Li 2007

11. RSA Key Size and Security
key size  size of n
n: 1024 bits (309 digits); bits (618 digits); 4096 bits
Security  difficulty of
Factorizing n=p*q p, q, φ, d
A 512 bits (154 digits) n could be factored in several months
© Yingjiu Li 2007

12. RSA Key Size and Security
By the year 2009, a machine that could break a 1024-bit RSA key in about a day would cost at least $250 million
For data that needs to be protected no later than the year 2015, the table indicates that the RSA key size should be at least 1024 bits. For data that needs to be protected longer, the key size should be at least 2048 bits.
© Yingjiu Li 2007

13. Asymmetric vs Symmetric
Key exchange over public channel
Scalable for multi-party communication
Long keys (e.g., 1024 bits)
Slow implementation
RSA software can encrypt 7.4~21.6 Kb/sec
Fastest RSA hardware can encrypt 1 Mb/sec
Symmetric
Key exchange must be done over secure channel
Non-scalable for multi-party communication
Relative short keys (e.g., 128 bits)
Fast implementation
In software, DES is generally 100 times faster than RSA
In hardware, DES is between 1000 to 10,000 times faster
© Yingjiu Li 2007

14. Scenario
RSA is too slow, not suitable for large P
© Yingjiu Li 2007

15. Question
Can we have a fast (as AES) and scalable crypto-system without secure channel (as RSA)?
Envelop encryption: combine AES and RSA
Es: AES encryption
Ds: AES decryption
k: AES key
Ea: RSA encryption
Da: RSA decryption
(e,n): RSA public key; d: RSA private key
© Yingjiu Li 2007

16. Envelop Encryption (1)
Alice selects a random session key (AES key) k, and gets C1=Es(k,P) (using AES)
Mallory
C1= Es(k,P)
Bob
Alice
© Yingjiu Li 2007

17. Envelop Encryption (2)
Alice uses Bob’s public key e,n to encrypt k (using RSA), and gets C2=Ea(e,n,k)
Mallory
C2=Ea(e,n,k) C1= Es(k,P)
Bob
Alice
© Yingjiu Li 2007

18. Communication Alice sends C1 and C2 together to Bob Mallory Bob Alice
C2=Ea(e,n,k) C1= Es(k,P)
Bob
Alice
© Yingjiu Li 2007

19. Envelop Decryption (1)
Bob uses his private key d to decrypt C2 (using RSA) and gets k = Da(d,C2)
Mallory
C2=Ea(e,n,k) C1= Es(k,P)
Bob
Alice
k=Da(d,C2)
© Yingjiu Li 2007

20. Envelop Decryption (2)
Bob uses k to decrypt C1 (using AES) and gets P=Ds(k,C1)
Mallory
C2=Ea(e,n,k) C1= Es(k,P)
Bob
Alice
k=Da(d,C2) P=Ds(k,C1)
© Yingjiu Li 2007

21. Demo Cryptool  Encrypt/Decrypt  Hybrid  RSA-AES encryption
RSA-AES decryption
© Yingjiu Li 2007

22. Hands-on Exercise Download Lab.doc
Generate RSA key pairs of 1024 and 2048 bits
Instruction 2.1
RSA encryption and decryption for small file and large file
Instruction 2.2
© Yingjiu Li 2007

23. Review When we say the key length of RSA is 1024 bits, we mean
n ) p,q ) e,d
Alice encrypts her message for Bob in RSA, which key should she use in encryption?
Alice’s private key
Alice’s public key
Bob’s public key
Alice encrypts her message for Bob in envelop encryption, which of the following is true?
Alice uses her private key to encrypt a session key, and uses the session key to encrypt her message
Alice uses her public key to encrypt a session key, and uses the session key to encrypt her message
Alice uses Bob’s public key to encrypt a session key, and uses the session key to encrypt her message
© Yingjiu Li 2007

24. Assignment 1
Submit hardcopy today
© Yingjiu Li 2007