1. David Evans http://www.cs.virginia.edu/evans
Lecture 2:
Perfect Ciphers
(in Theory, not Practice)
Shannon was the person who saw that the binary digit was the fundamental element in all of communication. That was really his discovery, and from it the whole communications revolution has sprung.
R G Gallager
I just wondered how things were put together.
Claude Shannon
Claude Shannon,
CS588: Cryptology
University of Virginia
Computer Science
David Evans

2. University of Virginia CS 588
Menu
Survey Results
Perfect Ciphers
Entropy and Unicity
25 January 2005
University of Virginia CS 588

3. Survey Responses
Majors: Cognitive Science, Computer Science (13), Computer Engineering (2), Economics, Math
Year: Third year: 6, Fourth Year: 10, Grad: 1, 2005: 1
Midterm: March 3: 12, March 15: 3, March 17: 2, March 22: 1
Broken in?: No: 13, Yes: 5
Full answers are on the course web site
25 January 2005
University of Virginia CS 588

4. What was the coolest thing you learned last semester?
“i've been sitting here for like five minutes trying to think of something. perhaps that aristotle (or perhaps plato) hates bands that sell out.”
“hmm.... I learned there was a huge molasses explosion at the Boston harbor in the early 20th century.... not really all that cool i guess because a lot of people died but still pretty interesting.”
25 January 2005
University of Virginia CS 588

5. University of Virginia CS 588
Break In Story
In high-school the administrators installed a fairly restrictive rights-control program called "FoolProof". It was so restrictive, however, that my AP Comp. Sci class was unable to compile and run programs on the main HD forcing us to use a floppy disk for our source and intermediate files, which was painfully slow as projects increased in size. Having unsuccessfully lobbied to have restrictions lifted so we could compile and run programs on a temporary directory. I tried to convince them that there were more eloquent and flexible security programs out there. So, I did a little research on the version of FoolProof being used. It turns out that this version of FoolProof had two major security flaws:
25 January 2005
University of Virginia CS 588

6. University of Virginia CS 588
Break In Story
1) A "backdoor" password used by tech support that is hardcoded into the executable and 2) The password information is stored in plaintext within main memory some number of bytes into one of its main data structures which was marked by an ascii string which was some sort of constant containing the version information.
So, I acquired a utility which would allow me to browse and search the contents of physical memory determined what this ascii string would be for the given version of FoolProof and searched memory. Sure enough I came up with a match and about 8 bytes later a plaintext ascii string:
"WhatAFool"
25 January 2005
University of Virginia CS 588

7. University of Virginia CS 588
Polling Protocol
25 January 2005
University of Virginia CS 588

8. University of Virginia CS 588
Last Time
Big keyspace is not necessarily a strong cipher
Claim: One-Time Pad is perfect cipher
In theory: depends on perfectly random key, secure key distribution, no reuse
In practice: usually ineffective (VENONA, Lorenz Machine)
Today: what does is mean to be a perfect cipher?
25 January 2005
University of Virginia CS 588

9. University of Virginia CS 588
Ways to Convince
“I tried really hard to break my cipher, but couldn’t. I’m a genius, so I’m sure no one else can break it either.”
“Lots of really smart people tried to break it, and couldn’t.”
Mathematical arguments – key size (dangerous!), statistical properties of ciphertext, depends on some provably (or believed) hard problem
Invulnerability to known cryptanalysis techniques (but what about undiscovered techniques?)
Show that ciphertext could match multiple reasonable plaintexts without knowing key
Simple monoalphabetic secure for about 10 letters of English: XBCF CF FWPHGW
This is secure
Spat at troner
25 January 2005
University of Virginia CS 588

10. University of Virginia CS 588
Claude Shannon
Master’s Thesis [1938] –
boolean algebra in electronic circuits
“Mathematical Theory of Communication” [1948] – established information theory
“Communication Theory of Secrecy Systems” [1945/1949] (linked from notes)
Invented rocket-powered Frisbee, could juggle four balls while riding unicycle
25 January 2005
University of Virginia CS 588

11. University of Virginia CS 588
Entropy
Amount of information in a message:
H(M) =  Prob [Mi] log ([1 / Prob [Mi])
over all possible messages Mi n
i=1
25 January 2005
University of Virginia CS 588

12. University of Virginia CS 588
Entropy
If there are n equally probable messages,
H(M) =  1/n log n
= n * (1/n log n))
= log n
Base of log is alphabet size, so for binary:
H(M) = log2 n
where n is the number of possible meanings n
i=1
25 January 2005
University of Virginia CS 588

13. Entropy Example M = months of the year H(M) = log2 12  3.6
(need 4 bits to encode a year)
25 January 2005
University of Virginia CS 588

14. Rate
Absolute rate: how much information can be encoded
R = log2 Z (Z=size of alphabet)
REnglish =
Actual rate of a language:
r = H(M) / N
M is a source of N-letter messages
r of months spelled out using 8-bit ASCII:
log2 26  4.7 bits / letter
= log2 12 / (8 letters * 8 bits/letter)  0.06
25 January 2005
University of Virginia CS 588

15. University of Virginia CS 588
Rate of English
r(English) is about .28 letters/letter
(1.3 bits/letter) [Book uses 1.5]
How can one estimate this?
How many meaningful 20-letter messages in English?
r = H(M) / N
.28 = H(M)/20
H(M) = 5.6 = log26 n
n = ~ 83 million (of 2*1028 possible)
Probability that 20-letters are sensible English is
About 1 in 2 * 1020
25 January 2005
University of Virginia CS 588

16. University of Virginia CS 588
Redundancy
Redundancy (D) is defined:
D = R – r
Redundancy in English:
D = = .72 letters/letter
D = 4.7 – 1.3 = 3.4 bits/letter
Each letter is 1.3 bits of content, and 3.4 bits of redundancy. (~72%)
7-bit ASCII (as in PS1, problem 5)
D = 7 – 1.3 = 5.7
81% redundancy, 19% information
25 January 2005
University of Virginia CS 588

17. University of Virginia CS 588
Unicity Distance
Entropy of cryptosystem: (K = number of possible keys)
H(K) = logAlphabet Size K
if all keys equally likely
H(64-bit key) = log2 264 = 64
Unicity distance is defined as:
U = H(K)/D
Expected minimum amount of ciphertext needed for brute-force attack to succeed.
25 January 2005
University of Virginia CS 588

18. University of Virginia CS 588
Unicity Examples
Monoalphabetic Substitution
H(K) = log2 26!  87
D = 3.4 (redundancy in English)
U = H(K)/D  25.5
Intuition: if you have 25 letters, probably only matches one possible plaintext.
D = 0 (random bit stream)
U = H(K)/D = infinite
25 January 2005
University of Virginia CS 588

19. University of Virginia CS 588
Unicity Distance
Probabilistic measure of how much ciphertext is needed to determine a unique plaintext
Does not indicate how much ciphertext is needed for cryptanalysis
If you have less than unicity distance ciphertext, can’t tell if guess is right.
As redundancy approaches 0, hard to cryptanalyze even simple cipher.
25 January 2005
University of Virginia CS 588

20. Shannon’s Theory [1945] Message space: { M1, M2,..., Mn }
Assume finite number of messages
Each message has probability
p(M1) + p(M2) p(Mn) = 1
Key space: { K1, K2,..., Kl }
(Based on Eli Biham’s notes)
25 January 2005
University of Virginia CS 588

21. Perfect Cipher: DefinitionKa
A perfect cipher: there is some key that maps any message to any ciphertext with equal probability.
For any i, j:
p (Mi|Cj) = p (Mi) M1 C1 Kb C2 M2
...
... Mn Cn
25 January 2005
University of Virginia CS 588

22. Conditional Probability
Prob [B | A] =
The probability of B, given that A occurs
Prob [ coin flip is tails ] = ½
Prob [ coin flip is tails | last coin flip was heads ]
= ½
Prob [ today is Monday | yesterday was Sunday]
= 1
Prob [ today is a weekend day | yesterday was a workday]
= 1/5
25 January 2005
University of Virginia CS 588

23. Calculating Conditional Probability
P [B | A ] = Prob [ A  B ]
Prob [ A ]
Prob [ coin flip is tails | last coin flip was heads ] =
Prob [ coin flip is tails and last coin flip was heads ]
Prob [ last coin flip was heads ]
= (½ * ½) / ½ = ½
25 January 2005
University of Virginia CS 588

24. P (today is a weekend day | yesterday was a workday)
= P (today is a weekend day and yesterday was a workday) / P (yesterday was a workday)
= P (today is a weekend day) * P(yesterday was a workday) / P (yesterday was a workday)
= 2/7 * 5/7 / 5/7 = 2/7
Wrong!
P(A  B) = P(A) * P(B)
only if A and B are independent events
25 January 2005
University of Virginia CS 588

25. University of Virginia CS 588
Perfect Cipher
Definition:  i, j: P (Mi|Cj) = P (Mi)
A cipher is perfect iff:
 M, C P (C | M) = P (C)
Or, equivalently:
 M, C P (M | C) = P (M)
25 January 2005
University of Virginia CS 588

26. University of Virginia CS 588
Perfect Cipher
 M, C P (C | M) = P (C)
 M, C P (C) =  P(K)
such that EK(M) = C
Or:
 C  P(K) is independent of M such that EK(M) = C K K
Without knowing anything about the key, any ciphertext is equally likely to match any plaintext.
25 January 2005
University of Virginia CS 588

27. Example: Monoalphabetic
Random monoalphabetic substitution for one letter message:
 C,M: P (C) = P (C | M) = 1/26.
25 January 2005
University of Virginia CS 588

28. University of Virginia CS 588
Example: One-Time Pad
For each bit:
P(Ci = 0) = P(Ci = 0 | Mi = 0) = P (Ci = 0 | Mi = 1) = ½
since Ci = Ki  Mi
P (Ki  Mi = 0) = P (Ki = 1) * P (Mi = 1)
+ P (Ki = 0) * P (Mi = 0)
Truly random K means P (Ki = 1) = P (Ki = 0) = ½
= ½ * P (Mi = 1) + ½ * P (Mi = 0)
= ½ * (P (Mi = 1) + P (Mi = 0)) = ½
All key bits are independent, so:
P(C) = P(C | M) QED.
25 January 2005
University of Virginia CS 588

29. Perfect Cipher Keyspace Theorem
Theorem: If a cipher is perfect, there must be at least as many keys (l) are there are possible messages (n).
25 January 2005
University of Virginia CS 588

30. Proof by Contradiction
Suppose there is a perfect cipher with
l < n. (More messages than keys.)
Let C0 be some ciphertext with P(C0) > 0.
There exist
m messages M such that M = DK(C0 )
n - m messages M0 such that M0  DK(C0 )
We know 1  m  l < n so n - m > 0 and there is at least one message M0.
25 January 2005
University of Virginia CS 588

31. University of Virginia CS 588
Proof, cont.
Consider the message M0 where
M0  DK(C0 ) for any K.
So,
P (C0 | M0) = 0.
In a perfect cipher,
P (C0 | M0) = P (C0) > 0.
Contradiction! It isn’t a perfect cipher.
Hence, all perfect ciphers must have l  n.
25 January 2005
University of Virginia CS 588

32. Example: Monoalphabetic
Random monoalphabetic substitution is not a perfect cipher for messages of up to 20 letters:
l = 26! n = 2620
l < n its not a perfect cipher.
In previous proof, could choose C0 = “AB” and M0 = “ee” and P (C0 | M0) = 0.
25 January 2005
University of Virginia CS 588

33. Example: Monoalphabetic
Is random monoalphabetic substitution a perfect cipher for messages of up to 2 letters?
l = 26! n = 262
l  n.
No! Showing l  n does not prove its perfect.
25 January 2005
University of Virginia CS 588

34. University of Virginia CS 588
Summary
Cipher is perfect:  i, j: p (Mi|Cj) = p (Mi)
Given any ciphertext, the probability that it matches any particular message is the same.
Equivalently,  i, j: p (Ci|Mj) = p (Ci)
Given any plaintext, the probability that it matches any particular ciphertext is the same.
25 January 2005
University of Virginia CS 588

35. University of Virginia CS 588
Imperfect Cipher
To prove a cipher is imperfect either:
Find a ciphertext that is more likely to be one message than another
Show that there are more messages than keys
Implies there is some ciphertext more likely to be one message than another even if you can’t find it.
25 January 2005
University of Virginia CS 588

36. University of Virginia CS 588
Charge
Problem Set 1: due Feb 3
All other questions covered (as much as we will cover them in class) already
Next time:
Classical ciphers (Nate Paul)
25 January 2005
University of Virginia CS 588