1. Computer Science 101 Cryptography Part II

2. Modern Cryptography
Enigma was last mechanical crypto device in widespread use
All modern crypto is done with digital electronics (i.e., computers)‏
Main applications
Military communication
Banking transactions
“Illicit” activity (terrorism, drugs, porn, file-sharing)‏

3. Symmetric vs. Asymmetric Crypto
Symmetric : Single key for both encryption and decryption (what we've seen thus far)‏
Both Alice and Bob must have the key
Best-known example: DES
Asymmetric : One key encryption, another for decryption
Bob publishes his public encryption key (Alice & Eve can both see it), hides his private decryption key
Best known examples: RSA

4. Symmetric Crypto
Block Cipher: Break digital message into small blocks (chunks) for encoding/transmission
E.g., using ASCII (8 bits per letter), 64-bit block can encode 8 characters
Basic idea: mix up (convolve, blend) block with key so much that each bit is “smeared” over whole block – like making dough from eggs, flower, water.
Then Eve has to extract eggs from dough!

5. Block Ciphers Plaintext is divided into blocks
Blocks are operated on to produce blocks in the ciphertext.
This obscures the letter structure of the plaintext
Each character of block contributes to all characters of the encrypted block.

6. Block Cipher - Example Matrix multiplication:
To obtain element in row I, column J, we multiply elements from row I of first matrix with corresponding elements of column J of second matrix and add up these products:

7. Block Cipher – Example (Cont.)
Matrix multiplication:
For this example, we now would reduce the numbers modulo 26

8. Block Cipher – Example (Cont.)
Now each character is given a numerical value.
For simplicity, let’s use position in alphabet A is 1, etc.
Each letter in plaintext is replaced by numerical value
Message “GENERALS” becomes
Again for simplicity, let’s use block size of 2
Message looks like

9. Block Cipher – Example (Cont.)
For block size of 2, we would encrypt using some 2 by 2 matrix, block size n uses n by n matrix.
We need an invertible matrix. Matrix A is invertible if there’s another matrix B so that multiplying A by B yields matrix with 1’s on diagonal and 0’s elsewhere.

10. Block Cipher – Example (Cont.)
So, we choose our invertible matrix.
To encrypt the message, we multiply each block times the encryption matrix to obtain encrypted block:
Message:
Becomes: or EXZGOVM from GENERALS
31 50
7 5
5 24
52 85
14 5
0 7
56 93
18 1
4 15
12 19

11. Block Cipher – Example (Cont.)
To decrypt, we reverse the process, using the inverse of the encryption matrix
Ciphertext EXZGOVM becomes:
Or GENERALS
33 -47
5 24
7 5
14 -21
0 7
14 5
18 -25
4 15
18 1
22 13

12. DES Stands for Data Encryption Standard
Designed to protect electronic information
A block cipher
Blocks: 64 bits long
Key: 64-bit binary key (only 56 bits are used)
Invitation to Computer Science, Java Version, Third Edition

13. DES (continued)
Every substitution, reduction, expansion, and permutation is determined by a well-known set of tables
The same algorithm serves as the decryption algorithm
Invitation to Computer Science, Java Version, Third Edition

14. Invitation to Computer Science, Java Version, Third Edition
Figure 13.11
The DES Encryption Algorithm
Invitation to Computer Science, Java Version, Third Edition

15. DES (continued) Triple DES Improves the security of DES
Requires two 56-bit keys
Runs the DES algorithm three times
AES (Advanced Encryption Standard)
Uses successive rounds of computations that mix up the data and the key
Key length: 128, 192, or 256 bits
Invitation to Computer Science, Java Version, Third Edition

16. DES: Trust Us, We're the Government
Data Encryption Standard : IBM (NSA?) block cipher
64-bit blocks
56 bit keys (64 bits, minus 8 for parity)‏
16 rounds
Hard to defeat using correlational techniques (c.f. Enigma)‏
Brute force : 256 = 72,057,600,000,000,000
possible keys to try
So we're safe, right?

17. DES: The Revenge of Moore's Law
Moore's Law: Computer power doubles every 18 months.
Each bit = factor of two
So to break N more bits in key, wait 18N months = 1.5N years
But DES was developed 30 years (20 bits) ago
Blaze, Diffie, Rivest, Schneier, et al. (1996) – 56 bits is no longer enough!

18. DES: The Revenge of Moore's Law
With parallel computing, you can “buy” more bits
First computer tries keys
Second computer tries keys
Of course, we have more than two computers (like our 48-node Beowulf cluster)!
Alternatives: 3DES (168 bits), Blowfish (448 bits),
Skipjack / Clipper (80 bits) – declassified 1998

19. Symmetric Key Crypto: General Issues
Problem: How do Alice and Bob share a key?
Alice sends key to Bob (but Eve sees it!)‏
Alice and Bob meet in secret
(inconvenient; impractical)‏
Alice has her key KA ; Bob has his KB ; and they send the message M back and forth; e.g.:
KA = 3
KB = 5
M = 4
Encrypt = *
Decrypt = /

20. Symmetric Crypto without Key Sharing12
A: 4*3 B

21. Symmetric Crypto without Key Sharing12
A: 4*3
B: 12*5

22. Symmetric Crypto without Key Sharing12
A: 4*3
B: 12*5 60

23. Symmetric Crypto without Key Sharing12
A: 4*3
B: 12*5 60
A: 60/3

24. Symmetric Crypto without Key Sharing12
A: 4*3
B: 12*5 60 60
A: 60/3 20

25. Symmetric Crypto without Key Sharing12
A: 4*3
B: 12*5 60 60
A: 60/3 20
B: 20/5

26. Symmetric Crypto without Key Sharing12
A: 4*3
B: 12*5 60 60
A: 60/3 20
B: 20/5 4

27. Symmetric Crypto without Key Sharing
Of course...
Keys are more complicated than 3, 5
Functions are more complicated that *, /
E.g., use block ciphers with 3DES, Blowfish, Skipjack, ...
Sending message three times is inefficient and may be costly
Can we avoid sharing secret keys and still send message only once?

28. The RSA Public-Key Cryptosystem
Rivest, Shamir, and Adelman, “A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, 21(2): , 1978
Asymmetric encryption algorithm – all senders to a given receiver use the same public key for encryption. Only the receiver has the private decryption key which is very different, but someway related.

29. Bob and Alice PA, SA public and secret keys for Alice
PB, SB public and secret keys for Bob
Alice
Bob
C= PA(M) SA PA M M

30. The Public and Private Keys
Select two large primes, p and q. (100 decimal digits, maybe)
Compute n = pq
Let m= (p-1)(q-1)
Select small positive integer e that shares no factors with m
Find d, which is the multiplicative inverse of e, modulo m (guaranteed to exist)
Publish the pair P = (e,n) as public key
Keep secret pair S = (d,n) as private key.

31. Encoding and Decoding Recall P=(e,n), S=(d,n)
Given message M, interpret as number in range 0 to (n-1).
Then encoding is P(M) = Me (modulo n) = C
Decoding is S(C) = Cd (modulo n) = (Me)d = Med = M all done modulo n.

32. RSA
Example: p = 5, q = 7, e = 5:
5 is a good value for d, because m = (p-1)(q-1) = 4*6 =24 and e*d = 5*5 = 25=1 (mod 24)
n = 5*7 = 35
To encode message M = 4:
C = M e mod n = 45 mod 35 = 1024 mod 35 = 9
To decode:
M = C d mod n = 95 mod 35 = mod 35 = 4

33. RSA Breaking the code means figuring out private key d.
Seems like Eve can do this easily, using algebra:
Make up arbitrary message M
Encode it using Bob's public key (e, n): C = M e mod n
Then solve M = C d mod n
For M = 4, n = 35, e = 5 example:
C = 9
4 = 9d mod 35

34. RSA Problem: Lots of values of d will work:
4 = 95 mod 35 = 911 mod 35 = 917 mod 35 = ...
In fact, Eve must know p and q if she wants to find d
(since Bob used p and q to generate d)‏
So the problem becomes factoring n into p*q.
So Bob uses huge p, q, and gets very huge n.
This is the “magic” of primes: there is no way other than exhaustive search.
The encoding function C = M e mod n is a trapdoor : easy to get in, difficult to get out!

35. What is this based on? Note: We make public (e,n), and keep secret d.
Note that an eavesdropper would only need to factor n, getting p and q and then could find d.
RSA is based on the fact that we do not have an algorithm that can factor large numbers in any feasible speed. Furthermore there is much evidence suggesting that there can not be such an algorithm.

37. PGP
Phil Zimmermann – politically active physicist / computer scientist; wants to make asymmetric (public-key) crypto available to ordinary citizens
But RSA algorithm runs too slow on 1980's PC's
Solution : combine symmetric (DES) and asymmetric (RSA) techniques

38. PGP: Pretty Good Privacy
Alice uses Bob's public RSA key to encrypt her secret DES key (small), and DES key to encrypt message (big). Sends both to Bob.
DESA A:
PubKeyB C1
C2C3...Cn M
DESA

39. PGP
Bob decrypts Alice's IDEA key using his private RSA key, then the message using A's IDEA key. B: C1
C2C3...Cn
PrivKeyB
DESA M

40. Secure Socket Layer
Figure 13.12
An SSL Session
Invitation to Computer Science, Java Version, Third Edition

41. Well gang, it's been fun, but now it's time to head
to the barn (sigh).