David Evanshttp://www.cs.virginia.edu/~evans

Lecture 13: Astrophysics and Cryptology

CS200: Computer ScienceUniversity of Virginia

Computer Science

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Menu

• Quicksort Recap

• DeGrasse Tyson’s Essay

• Cryptography (CS588 condensed)

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Quicksort

C. A. R. Hoare, 1961

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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))))))

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filter

(define (filter f lst) (insertlg (lambda (el rest)

(if (f el) (cons el rest) rest)) lst null)) How much work is filter?

(n)

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Quicksort

• filter is (n)• How much work is Quicksort if the input

list is sorted?

Worst Case: (n2)we filter n times, each is (n)

(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))))))

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Quicksort

• 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 listWe need log2 n splits to get down to empty listBest (Average) Case: (n log2 n)

we filter log2 n times, each is (n)

(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))))))

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> (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

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0

2000

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66

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n log2 n(quicksort)

n2 (bubblesort)

Growth of time to sort random list

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Science’s Endless Golden Age

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Astrophysics• “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)?

(n3)When we double the size of the simulation, the work octuples! (Just like oceanography octopi simulations)

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Astrophysics and Moore’s Law• Simulating universe is (n3)• 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?

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(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)))) (knowledge-of-universe (find-biggest-scale 1)))

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> (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

Will there be any mystery left in the Universe when you die?

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Liberal Arts• Grammar: study of meaning in written

expression• Rhetoric: comprehension of verbal

and written discourse• Logic: argumentative discourse for

discovering truth• Arithmetic: understanding numbers• Geometry: quantification of space• Music: number in time• Astronomy: laws of the planets and

stars

Yes, we need to understandmeaning to describe

computations

Interfaces between components, discourse

between programs and users

Logic for controlling and reasoning about

computations

Yes (last few lectures)

Yes (PS 1, 2, 3)

Yes, its called GEB for a reason!

No, but astronomy uses CS a lot.

Triv

ium

Qua

driv

ium

Correction from Lecture 1:

Yes (Neil DeGrasses Tyson says so!)

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Bold (Possibly Untrue) Claim

This course is the most consistent with the original intent of a Liberal Arts education of any course offered at UVA this semester!

Correction from Lecture 1:

since Mr. Jefferson founded it!

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The Endless Golden Age

• 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.

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The Real Golden Rule?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?

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Cryptology(CS588 Condensed)

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Terminology

Encrypt DecryptPlaintextCiphertext

Plaintext

Alice Bob

Eve

Insecure Channel

C = E(P)P = D(C)E must be invertible: P = D (E (P))

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Encrypt DecryptPlaintextCiphertext

Plaintext

Alice Bob

Insecure Channel

C = E(P, K)P = D(C, K)

K K

“The enemy knows the system being used.”

Claude Shannon

Eve

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Jefferson Wheel Cipher

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Enigma• 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

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EnigmaComing to US in April!

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Bletchley Park

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Lorenz Cipher Machine

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Perfectly Secure Cipher: One-Time Pad

• Mauborgne/Vernam [1917]• xor ():

0 0 = 0 1 0 = 10 1 = 1 1 1 = 0a a = 0a 0 = aa b b = a

• E(P, K) = P KD(C, K) = C K = (P K) K = P

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For any given ciphertext, all plaintexts are equally possible.

Ciphertext: 0100111110101

Key: 1100000100110

Plaintext: 1000111010011 = “CS”

Why perfectly secure?

1

0 B

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If its “perfect” why is it broken?

• Cannot reuse K

• Need to generate truly random bit sequence as long as all messages

• Need to securely distribute key

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“One-Time” Pad’s in Practice• 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

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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

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From http://www.codesandciphers.org.uk/lorenz/fish.htm

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Problem Set 4

• 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!

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Motivation Helps…

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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Modern Ciphers

• 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

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Charge

• 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)