1. Class 20: Running Time
cs1120 Fall 2011
David Evans
7 October 2011

2. Image by Jonathan Mak 2

3. 3

4. Next House, MIT
4

5. Whole Earth Epilog, 1974
“On the back cover of their
final issue was a
photograph of an early
morning country road, the
kind you might find yourself
hitchhiking on if you were
so adventurous. Beneath it
were the words: “Stay
Hungry. Stay Foolish.” It
was their farewell message
as they signed off. Stay
Hungry. Stay Foolish. And I
have always wished that
for myself.”
Steve Job’s 2005 Stanford
Commencement Speech
5

6. Turing Machines
One of the deep mysteries to me is our logo, the
symbol of lust and knowledge, bitten into, all
crossed with the colors of the rainbow in the wrong
order. You couldn’t dream a more appropriate logo:
lust, knowledge, hope and anarchy.
Jean-Louis Gassée
6

7. Last Class: TM for Unary >
Input: 1a>1b#
Output: #1 if a > b, #0 otherwise.
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8. Transition Rules
(CrossOne, 1) -> (LookForGT, X, R) ; found a 1 in a, look for a 1 in b to cross off
(CrossOne, >) -> (Write0Out, #, R) ; no more 1's in a, so a <= b
(LookForGT, 1) -> (LookForGT, 1, R) ; keep looking
(LookForGT, >) -> (LookForB1, >, R) ; found the >
(LookForB1, X) -> (LookForB1, X, R) ; keep looking
(LookForB1, 1) -> (GoBack, X, L) ; crossed off match, return for next
(LookForB1, #) -> (Write1Out, #, R) ; no match, a > b
(GoBack, X) -> (GoBack, X, L) ; keep looking for >
(GoBack, >) -> (Reset, >, L) ; now, look for the rightmost X
(Reset, 1) -> (Reset, 1, L) ; keep looking for X
(Reset, X) -> (CrossOne, X, R) ; found it!
(Write0Out, *) -> (Halt, 0, HALT)
(Write1Out, *) -> (Halt, 1, HALT)
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9. (CrossOne, 1) -> (LookForGT, X, R) (GoBack, X) -> (GoBack, X, L)
(CrossOne, >) -> (Write0Out, #, R) (GoBack, >) -> (Reset, >, L)
(LookForGT, 1) -> (LookForGT, 1, R) (Reset, 1) -> (Reset, 1, L)
(LookForGT, >) -> (LookForB1, >, R) (Reset, X) -> (CrossOne, X, R)
(LookForB1, X) -> (LookForB1, X, R) 1/1, (Write0Out, *) -> (Halt, 0, HALT)
(LookForB1, 1) -> (GoBack, X, L) (Write1Out, *) -> (Halt, 1, HALT)
(LookForB1, #) -> (Write1Out, #, R)
Look
1/X, X/X,
For Look
GT >/>, For
Cross B1
Start #/#,
One 1/X,
>/>,
Reset Go
>/#, X/X, Back
Write
1 Out
Write
1/1, X/X,
0 Out
/0,Halt /1,Halt
HALT
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10. Binary >
Input: a>b#
Output: #1 if the value a interpreted as a
binary number is greater than the value of b
interpreted as a binary number, #0 otherwise.
10

11. 1/1,
Look
1/X, X/X,
For Look
GT >/>, For
Cross B1
Start #/#,
One 1/X,
>/>,
Reset Go
>/#, X/X, Back Write
1 Out
Write
1/1, X/X,
0 Out
/0,Halt /1,Halt
HALT
11

12. 1/1,
0/0,
X/X,
Look Look
For >/>, For #/#,
GT-0 B-0
1/1, 1/X,
0/0, 0/X,
0/X,
Look
1/X, X/X,
For Look
GT >/>, For
Cross B1
Start #/#,
One
>/>, 1/X,
Reset Go
>/#, X/X, Back Write
1 Out
Write
1/1, X/X, /1,Halt
0 Out
/0,Halt Lots more changes needed…
but you get the idea
HALT
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13. “Sensible” Algorithm Analysis
Writing TM programs is long and tedious
(and, at least for normal size inputs, does not
correspond well to what a real machine does for
simple operations)
Common Simplifying Assumptions
Arithmetic operations are constant time
This is true if numbers are bounded
Cannot be true for arbitrarily large numbers
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14. What’s Coming Next
Chapter 7: Cost
Chapter 8: Searching and Sorting
We’ll do lots of examples in class also
Chapter 9: Mutation (PS5)
Chapter 10: Objects (PS6)
Ways to program more expressively
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15. Breaking Lorenz
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16. One-Time Pad
C=M K
Claude Shannon, A Mathematical Theory of
Cryptography, 1946 (declassified later): Proved
that a one-time pad is a perfect cipher! Frank Miller 1882
PS4, Question 3!
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17. Why perfectly secure?
For any given ciphertext, all plaintexts are
equally possible.
Ciphertext: J = 01001
Key1: I = 00110
Plaintext1: 01111 = K
Key2: L = 10010
Plaintext2: = 11011 = shift

18. Vernam’s One-Time Pad (1919)
A long paper tape with random
letters on it (using Baudot
code)
Cannot reuse key – tape must be
very long!
This has 6 holes per letter
(not Baudot code)

19. Lorenz Cipher
Based on the Vernam and Morehouse machines
Used Baudot code
Believed managing long paper tapes during
wartime was too difficult
Machine generates key sequence
If two machines start in same configuration, same key
sequence
Will not repeat for ~ 1019 letters
All words ever written by all humans is 1018 letters

21. HQIBPEXEZMUG!
May 1941: GCHQ learns about first
Fish link (Tunny)
Intercepted unencrypted
Baudot-encoded test messages
August 30, 1941: GCHQ Today
(not what it looked like in 1941!)
Operator retransmits failed message
with same starting configuration
Gets lazy and uses some
abbreviations, makes some SPRUCHNUMMER/SPRUCHNR
(Serial Number)
mistakes
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22. “Two Time” Pad
Allies have intercepted:
C1 = M1 K1
C2 = M2 K1
Same key used for both (same starting configuration)
22

23. “Cribs”
Know: C1, C2 (intercepted ciphertext)
C1 C2 = M1 M2
Don’t know M1 or M2
Can make some guesses (cribs)
SPRUCHNUMMER
Sometimes allies moved ships, sent out bombers to help the
cryptographers get good cribs
Given guess for M1, calculate M2
M2 = C1 C2 M1
If M2 seems plausible, calculate key:
K1 = M1 C1 = M2 C2
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24. Learning the Key
From the 2 intercepted
messages, Col. John Tiltman
worked on guessing cribs to
find M1 and M2
4000 letter messages
Found 4000 letter key K1
24

25. Reverse Engineering Lorenz
Major G. W. Morgan to Bill Tutte
(recent Chemistry graduate):
“See what you can do with
this.” (Tiltman’s K1)
Found period of 574 = 41 14
Six months later new machine
structure likely to generate K1
Bill Tutte
U. Waterloo
(1917-2002)
25

26. Lorenz Structure
Main weakness: each step, either all
S wheels turn, or none of them do!
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27. Intercepting Traffic
Set up listening post to intercept traffic from 12
Lorenz (Fish) links
Different links between Berlin and conquered capitals
Slightly different coding procedures, and different
configurations
600 people worked
on intercepting traffic
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28. Breaking Traffic
Knew machine structure, but a different initial
configuration was used for each message
Need to determine wheel setting:
Initial position of each of the 12 wheels
Needed to try them fast enough to decrypt message
while it was still strategically valuable
This is what you did for PS4 (except with fewer wheels)
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29. Recognizing a Good Guess
Intercepted Message (divided into 5 channels for
each Baudot code bit)
Zc = z0z1z2z3z4z5z6z7…
zc, i = mc,i xc,i sc,i
Message Key (parts from S-wheels and rest)
Look for statistical properties
How many of the zc,i’s are 0? ½ (not useful)
How many of (zc,i+1 zc,i) are 0? ½
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30. Double Delta
Zc,i = Zc,i Zc,i+1
Combine two channels:
Z1,i Z2,i = M1,i M2,i > ½ Yippee!
X1,i X2,i = ½ (key)
S1,i S2,i > ½ Yippee!
Why is M1,i M2,i > ½
Message is in German, more likely following
letter is a repetition than random
Why is S1,i S2,i > ½
S-wheels only turn when M-wheel is 1
30

31. Actual Advantage
Probability of repeating letters
Prob[ M1,i M2,i = 0] 0.614
3.3% of German digraphs are repeating
Probability of repeating S-keys
Prob[ S1,i S2,i = 0] ~ 0.73
Prob[ Z1,i Z2,i X1,i X2,i = 0]
= 0.614 * 0.73 + (1-0.614) * (1-0.73)
M and S are 0 M and S are 1
= 0.55 if the wheel settings guess is correct (0.5 otherwise)
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32. Using the Advantage
If the guess of X is correct, should see higher than ½
of the double deltas are 0
Try guessing different configurations to find highest
number of 0 double deltas
Problem:
# of double delta operations to try one config
= length of Z * length of X
= for 10,000 letter message
1271 settings * 7 per double delta
= 89 M operations
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33. Heath Robinson
Dec 1942: Decide to build a
machine to do these s
quickly, due June 1943
Apr 1943: first Heath Robinson
machine is delivered!
Intercepted ciphertext on tape:
2000 characters per second (12
miles per hour)
Needed to perform 7 operations
each ½ ms
Heath Robinson, British Cartoonist (1872-1944)
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34. Colossus
Heath Robinson machines were too slow
Colossus designed and first built in Jan 1944
Replaced keytext tape loop with electronic keytext
generator
Speed up ciphertext tape:
5,000 chars per second = 30 mph
Perform 5 double deltas simultaneously
Speedup = 2.5X for faster tape 5X for parallelism
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35. Colossus Design Ciphertext Tape
Electronic
Keytext Logic Tape Reader
Generator
Counter Position Counter
Printer
35

36. Impact on WWII
10 Colossus machines operated at Bletchley
Decoded 63 million letters in Nazi messages
Learned German troop locations to plan D-Day
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37. Colossus History
Kept secret after the war, all machines destroyed
During WWII Rebuilt (Summer 2004)
37

38. How could Bletchley Park solve a problem
a quintillion times harder than PS4?
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39. Poster in RAF Museum
39

40. Confronted with the prospect
of defeat, the Allied
cryptanalysts had worked night
and day to penetrate German
ciphers. It would appear that
fear was the main driving force,
and that adversity is one of the
foundations of successful
codebreaking.
Simon Singh, The Code Book
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41. Charge
No problem on your exam is one quintillionth as hard
as breaking Lorenz!
No real fear or adversity should be necessary
to solve the exam problems.
Exam 1: Out Now
Due at beginning of class Wednesday
Stay Hungry! Stay Foolish!
Watch or read Steve Jobs’ Commencement Speech
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