CS200: Computer Science
University of Virginia
Computer Science
David Evans http://www.cs.virginia.edu/~evans

(CS588 condensed)
13 February 2002 CS 200 Spring 2002 2

C. A. R. Hoare, 1961
13 February 2002 CS 200 Spring 2002 3

(define (quicksort cf lst)
(if (null? lst) lst
(append
(quicksort cf
(filter (lambda (el) (cf el (car lst)))
(cdr lst)))
(list (car lst))
(quicksort cf
(filter (lambda (el) (not (cf el (car lst))))
(cdr lst))))))
13 February 2002 CS 200 Spring 2002 4

(define (filter f lst)
(insertlg
(lambda (el rest)
(if (f el) (cons el rest) rest)) lst null))
How much work is filter?

n
13 February 2002 CS 200 Spring 2002 5

(define (quicksort cf lst)
(if (null? lst) lst
(append
(quicksort cf
(filter (lambda (el) (cf el (car lst)))
(cdr lst)))
(list (car lst))
(quicksort cf
(filter (lambda (el) (not (cf el (car lst))))
(cdr lst))))))
• filter is

( n )
• How much work is Quicksort if the input list is sorted?
Worst Case:

( n 2 ) we filter n times, each is

( n )
13 February 2002 CS 200 Spring 2002 6

Quicksort
(define (quicksort cf lst)
(if (null? lst) lst
(append (quicksort cf
(filter (lambda (el) (cf el (car lst))) (cdr lst)))
(list (car lst))
(quicksort cf
(filter (lambda (el) (not (cf el (car lst)))) (cdr lst))))))
• filter is

( n )
• How much work is Quicksort if the input list is random?
Each time we split the list, each piece is approximately ½ the length of the original list
We need log
Best (Average) Case:

( n log we filter
2 n splits to get down to empty list log
2
2 n ) n times, each is

( n )
13 February 2002 CS 200 Spring 2002 7

> (define r1000 (rand-int-list 1000))
> (time (sort < r1000)) cpu time: 1372 real time: 1372 gc time: 0
> (time (quicksort < r1000)) cpu time: 71 real time: 70 gc time: 0
> (define r2000 (rand-int-list 2000))
> (time (sort < r2000)) cpu time: 5909 real time: 5909 gc time: 0
> (time (quicksort < r2000)) cpu time: 180 real time: 180 gc time: 0
> (time (quicksort < (revintsto 1000))) cpu time: 2684 real time: 2684 gc time: 0
13 February 2002 CS 200 Spring 2002 8

12000
10000
8000
6000
4000
2000
0
13 February 2002
Growth of time to sort random list
CS 200 Spring 2002 n 2
(bubblesort) n log
2 n
(quicksort)
9

13 February 2002 CS 200 Spring 2002 10

• “If you’re going to use your computer to simulate some phenomenon in the universe, then it only becomes interesting if you change the scale of that phenomenon by at least a factor of 10. … For a 3D simulation, an increase by a factor of 10 in each of the three dimensions increases your volume by a factor of 1000.”
• How much work is astrophysics simulation (in  notation)?

( n 3 )
When we double the size of the simulation, the work octuples!
(Just like oceanography octopi simulations)
13 February 2002 CS 200 Spring 2002 11

• Simulating universe is

n 3
• Moore’s law: computing power doubles every 18 months
• Tyson: to understand something new about the universe, need to scale by 10x
• How long does it take to know twice as much about the universe?
13 February 2002 CS 200 Spring 2002 12

(define (computing-power nyears)
(if (= nyears 0) 1 (* 1.587 (computing-power (- nyears 1)))))
;;; doubling every 18 months = ~1.587 * every 12 months
(define (simulation-work scale) (* scale scale scale))
;;; Simulation is O(n^3) work
(define (log10 x) (/ (log x) (log 10)))
;;; primitive log is natural (base e)
(define (knowledge-of-universe scale) (log10 scale))
;;; knowledge of the universe is log 10 the scale of universe we can simulate
(define (find-knowledge-of-universe nyears)
(define (find-biggest-scale scale)
; today, can simulate size 10 universe
(if (> (/ (simulation-work scale) 1000)
(computing-power nyears))
(- scale 1)
(find-biggest-scale (+ scale 1))))
CS 200 Spring 2002 13

> (find-knowledge-of-universe 0)
1.0
> (find-knowledge-of-universe 1)
1.041392685158225
> (find-knowledge-of-universe 2)
1.1139433523068367
> (find-knowledge-of-universe 5)
1.322219294733919
> (find-knowledge-of-universe 10)
1.6627578316815739
> (find-knowledge-of-universe 15)
2.0
> (find-knowledge-of-universe 30)
3.00560944536028
> (find-knowledge-of-universe 60)
5.0115366121349325
> (find-knowledge-of-universe 80)
6.348717927935257
13 February 2002
Will there be any mystery left in the Universe when you die?
CS 200 Spring 2002 14

Correction from Lecture 1:
• Grammar: study of meaning in written expression
Yes, we need to understand meaning to describe computations
• Rhetoric: comprehension of verbal and written discourse
• Logic: argumentative discourse for components, discourse between programs and users
Logic for controlling discovering truth and reasoning about computations
• Arithmetic: understanding numbers
Yes (last few lectures)
• Geometry: quantification of space
Yes (PS 1, 2, 3)
• Music: number in time
Yes, its called GE B
• Astronomy: laws of the planets and for a reason!
stars
Interfaces between
13 February 2002 CS 200 Spring 2002 15

Correction from Lecture 1:
since Mr. Jefferson founded it!
13 February 2002 CS 200 Spring 2002 16

• Golden Age – period in which knowledge/quality of something doubles quickly
• At any point in history, half of what is known about astrophysics was discovered in the previous 15 years!
• Moore’s law today, but other advances previously: telescopes, photocopiers, clocks, etc.
13 February 2002 CS 200 Spring 2002 17

Why do fields like astrophysics, medicine, biology and computer science (?) have “endless golden ages”, but fields like
– music (1775-1825)
– rock n’ roll (1962-1973, or whatever was popular when you were 16)
– philosophy (400BC-350BC?)
– art (1875-1925?)
– soccer (1950-1974)
– baseball (1925-1950)
– movies (1930-1940) have short golden ages?
What about mathematics?
13 February 2002 CS 200 Spring 2002 18

( CS588 Condensed)
13 February 2002 CS 200 Spring 2002 19

Plaintext Encrypt
Insecure Channel
Ciphertext
Decrypt Plaintext
Alice
13 February 2002
Eve
C = E(P)
P = D(C)
E must be invertible: P = D (E (P))
CS 200 Spring 2002
Bob
20

Plaintext
“The enemy knows the system being used.”
Claude Shannon
Encrypt
Insecure Channel
Ciphertext
Decrypt Plaintext
Alice
13 February 2002
K
Eve
C = E(P, K)
P = D(C, K)
CS 200 Spring 2002
K
Bob
21

13 February 2002
Jefferson Wheel Cipher
CS 200 Spring 2002 22

13 February 2002
• About 50,000 used by Nazi’s in
WWII
• Modified throughout WWII, believed to be perfectly secure
• Broken by Bletchley Park led by Alan Turing (and 30,000 others)
• First computer (Collossus) developed to break Nazi codes
(but kept secret through 1970s)
• Allies used decrypted Enigma messages to plan D-Day
CS 200 Spring 2002 23

Coming to US in April!
13 February 2002 CS 200 Spring 2002 24

13 February 2002 CS 200 Spring 2002 25

13 February 2002 CS 200 Spring 2002 26

• Mauborgne/Vernam [1917]
• xor ( 
):
0

0 = 0 1

0 = 1
0

1 = 1 1

1 = 0 a
 a = 0 a

0 = a a
 b
 b = a
• E(P, K) = P 
K
D(C, K) = C

K = (P

K)

K = P
13 February 2002 CS 200 Spring 2002 27

For any given ciphertext, all plaintexts are equally possible.
Ciphertext: 0100111110101
Key: 1100000100110
Plaintext:
13 February 2002 CS 200 Spring 2002 28

• Cannot reuse K
• Need to generate truly random bit sequence as long as all messages
• Need to securely distribute key
13 February 2002 CS 200 Spring 2002 29

• Lorenz Machine –
Nazi high command in WWII
– Pad generated by 12 rotors
– Receiver and sender set up rotors in same positions
– One operator retransmitted a message (but abbreviated message header the second time!)
– Enough for Bletchley Park to figure out key – and structure of machine that generated it!
– But still had to try all configurations
13 February 2002 CS 200 Spring 2002 30

Colossus – First
Programmable
Computer
• Bletchley Park, 1944
• Read ciphertext and
Lorenz wheel patterns from tapes
• Tried each alignment, calculated correlation with
German
• Decoded messages (63M letters by 10 Colossus machines) that enabled Allies to know German troop locations to plan D-Day
• Destroyed in 1960, kept secret until 1970s
13 February 2002 CS 200 Spring 2002 31

13 February 2002
From http://www.codesandciphers.org.uk/lorenz/fish.htm
CS 200 Spring 2002 32

• Break a simplified Lorenz Cipher
• Removed one wheel, made initial positions of all groups of wheels have to match
• Small rotors
• Its REALLY AMAZING that the British were able to break the real Lorenz in 1943 and it is still hard for us today!
13 February 2002 CS 200 Spring 2002 33

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
13 February 2002 CS 200 Spring 2002 34

• 128-bit keys, encrypt 128-bit blocks
• Brute force attack
– Try 1 Trillion keys per second
– Would take 10790283070806000000 years to try all keys!
– If that’s not enough, can use 256-bit key
• No known techniques that do better than brute force search
13 February 2002 CS 200 Spring 2002 35

• PS4
– No new Computer Science concepts
– You should be able to do it now
– Lots of practice with lists and recursion
– No bombs dropping, but a little bit of motivation
(prize for first person/group to decipher secret message)
13 February 2002 CS 200 Spring 2002 36