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

CS588: Security and PrivacyUniversity of VirginiaComputer Science

Lecture 5: Enigma Concluded

Bletchley Park (June 2004)

3 February 2005 University of Virginia CS 588 2

Last Time: Added Plugboard

Plaintext

B

Plugboard

L

Rotor1

M

Rotor2

N

Rotor3

R

ReflectorCiphertext

6 plugs: (26*25)/2 * (24*23)/2 * … * (16*15/2) / 6! = 1011 times more keys

3 February 2005 University of Virginia CS 588 3Poster in RAF Museum

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Operation

• Day key (distributed in code book)• Each message begins with message

key (“randomly” chosen by sender) encoded using day key

• Message key sent twice to check• After receiving message key, re-orient

rotors according to key

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Letter PermutationsSymmetry of Enigma:

if Epos (x) = y we know Epos (y) = xGiven message openingsDMQ VBM E1(m1) = D E4(m1) = V E1oE4(D) = VVON PUY => E1(D) = m1

PUC FMQ => E4 (E1 (D)) = V With enough message openings, we can build complete cycles for each position pair:

E1oE4 = (DVPFKXGZYO) (EIJMUNQLHT) (BC) (RW) (A) (S)Note: Cycles must come in pairs of equal length

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Composing Involutions• E1 and E2 are involutions (x y y x)• Without loss of generality, we can write:

E1 contains (a1a2) (a3a4) … (a2k-1a2k) E2 contains (a2a3) (a4a5) … (a2ka1) E1 E2

a1 a2 a2 x = a3 or x = a1

a3 a4 a4 x = a5 or x = a1Why can’tx be a2 or a3?

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Rejewski’s TheoremE1 contains (a1a2) (a3a4) … (a2k-1a2k)

E4 contains (a2a3) (a4a5) … (a2ka1)

E1E4 contains (a1a3a5…a2k-1)

(a2ka2k-2… a4a2)• The composition of two involutions consists of

pairs of cycles of the same length• For cycles of length n, there are n possible

factorizations

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Factoring PermutationsE1E4 = (DVPFKXGZYO) (EIJMUNQLHT) (BC) (RW)

(A) (S)

(A) (S) = (AS) o (SA) (BC) (RW) = (BR)(CW) o (BW)(CR) or = (BW)(RC) o (WC) (BR)(DVPFKXGZYO) (EIJMUNQLHT)

= (DE)(VI)… or (DI)(VJ) … or (DJ)(VM) … … (DT)(VE) 10 possibilities

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How many factorizations?(DVPFKXGZYO) (EIJMUNQLHT)

D a2

E1 E2

a2 VV a4 a4 P

Once we guess a2 everything else must follow!So, only n possible factorizations for an n-letter cycle

Total to try = 2 * 10 = 20

E2E5 and E3E6 likely to have about 20 to try also

About 203 (8000) factorizations to try

(still too many in pre-computer days)

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Luckily…• Operators picked message keys

(“cillies”)– Identical letters– Easy to type (e.g., QWE)

• If we can guess P1 = P2 = P3 (or known relationships) can reduce number of possible factorizations

• If we’re lucky – this leads to E1 …E6

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Solving?

E1 = B-1L-1Q LB

E2 = B-1L-2QL2B

E3 = B-1L-3QL3B

E4 = B-1L-4QL4B

E5 = B-1L-5QL5B

E6 = B-1L-6QL6B

6 equations, 3 unknowns

Not known to be efficiently solvable

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Solving?E1 = B-1L-1Q LB

BE1B-1 = L-1Q L

6 equations, 2 unknowns – solvable

Often, know plugboard settings (didn’t change frequently)

6 possible arrangements of 3 rotors, 263 starting locations= 105,456 possibilitiesPoles spent a year building a catalog of cycle structurescovering all of them (until Nov 1937): 20 mins to breakThen Germans changed reflector and they had to start over.

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1939• Early 1939 – Germany changes scamblers and

adds extra plugboard cables, stop double-transmissions– Poland unable to cryptanalyze

• 25 July 1939 – Rejewski invites French and British cryptographers– Gives England replica Enigma machine constructed

from plans, cryptanalysis• 1 Sept 1939 – Germany invades Poland, WWII

starts

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Bletchley Park• Alan Turing leads British effort to crack

Enigma• Use cribs (“WETTER” transmitted every day

at 6am) to find structure of plugboard settings• Built “bombes” to automate testing• 10,000 people worked at Bletchley Park on

breaking Enigma (100,000 for Manhattan Project)

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Alan Turing’s “Bombe”

Steps through all possible rotor positions (263), testing for probable plaintext; couldn’t search all plugboard settings (> 1012); take advantage of loops in cribs

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Enigma Cryptanalysis• Relied on combination of sheer brilliance,

mathematics, espionage, operator errors, and hard work

• Huge impact on WWII– Britain knew where German U-boats were– Advance notice of bombing raids– But...keeping code break secret more important

than short-term uses• The Coventry bombing story isn’t true, but decoy scouts is

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Projects• Start thinking about projects and forming

teams• Prefer teams of 3 people

– In rare circumstances will allow solo projects• Preliminary Proposals due Feb 15

– List of team members• Use web forum to discuss project ideas, find interested

teammates– Either:

• Short blurbs for at least 3 project ideas• A description of a project idea with some background,

convincing argument it will be possible and interesting

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Project Option 1: Research• Research Projects

– Identify an interesting problem related to cryptography/security

• Devise an approach for solving it– Analyze security of some system

• GET PERMISSION FIRST!• Doesn’t need to be limited to purely

computational systems

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Project Option 2: “Outreach”• Do something relevant to this course that

is beneficial to the larger community. Examples include: – Develop and teach a course for K-12 students

that uses cryptography to make math interesting

– Produce something that conveys important security principles to the general public

• Movie screening at end of class today

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Which should you do?

• Grad student, grad-school bound 3rd or 4th year: research project that integrates with your main research

• 3rd/4th year looking for a job: something you can talk about on job interviews

• 4th year with a job: outreach project• 4th year who doesn’t want a job

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Modern Symmetric Ciphers

A billion billion is a large number, but it's not that large a number.— Whitfield Diffie

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Goals of Cipher:Diffusion and Confusion

• Claude Shannon [1945]• Diffussion:

– Small change in plaintext, changes lots of ciphertext– Statistical properties of plaintext hidden in ciphertext

• Confusion:– Statistical relationship between key and ciphertext as

complex as possible• So, need to design functions that produce output

that is diffuse and confused

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Block Ciphers

• Stream Ciphers– Encrypts small (bit or byte) units one at a time

• Block Ciphers– Encrypts large chunks (64 bits) at once

• Ciphers we have seen so far: – Changing one letter of message only changes one

letter of ciphertext– There were classical ciphers that had some

diffusion: Vigenère autokey, Hill cipher (2-letter chunks)

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Ideal Block Cipher• 64 bit blocks• 264 possible plaintext blocks, must have at

least 264 corresponding ciphertext blocks – There are 264! possible mappings

• Why not just create a random mapping?– Need a 264 * 64-bit table 1021 bits – $14 quadrillion– Need to distribute new table if compromised

• Approximate ideal random mapping using components controlled by a key

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Feistel Cipher StructurePlaintext

Rou

nd

L0 R0

F

K1

L1 R1

L0 = left half of plaintextR0 = right half of plaintext

Li = Ri - 1 Ri = Li - 1 F (Ri - 1, Ki )

C = Rn || Ln

n is number of rounds (undo last permutation)

Sub

stitu

tion

Per

mut

atio

n

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One Round Feistel

E (L0 || R0):

L1 = R0

R1 = L0 F (R0, K1))

C = R1 || L1 = L0 F (R0, K1)) || R0

Li = Ri - 1

Ri = Li - 1 F (Ri - 1, Ki )

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DecryptionCiphertext

LD0RD0

F

Kn

L1 R1

LD0 = left half of ciphertextRD0 = right half of ciphertext

LDi = RDi - 1 RDi = LDi - 1

F (RDi - 1, Kn – i + 1)

P = RDn || LDn

n is number of rounds

Sub

stitu

tion

Per

mut

atio

n

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Decryption

D (L0 F (R0, K1)) || R0)LD0 = L0 F (R0, K1) RD0 = R0

LD1 = R0

RD1 = LD0 F (RD0, K1)= L0 F (R0, K1) F (RD0, K1)) = L0

P = RD1 || LD1 = L0 || R0 Yippee!

LDi = RDi - 1

RDi = LDi - 1 F (RDi - 1, Kn – i + 1)

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Multiple Rounds• The entire round is a function:

fK (L || R) = R || L F (R, K)) swap (L || R) = R || L

• E = swap ° swap ° fKr ° swap ° fKr-1

°

... ° fK2 ° swap ° fK1

• D = fK1 ° swap ° fK2

° ... °

fKr-1 ° swap ° fKr ° swap ° swap

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Decryption

swap (fK (swap (fK (L || R))= swap (fK (swap (R || L F (R, K))))

= swap (fK (L F (R, K) || R))

= swap (R || (L F (R, K)) F (R, K))= swap (R || L) = L || R

So swap ° fK its own inverse!

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F

• What are the requirements on F?– For decryption to work: none!– For security:

• Hide patterns in plaintext• Hide patterns in key• Coming up with a good F is hard

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Charge• Start to think of interesting project ideas• Post on discussion forum, find

teammates• Next time:

– DES: a Feistel cipher– Breaking DES (including Girish Ratanpal)

• Movie (if you can stay, otherwise it is on the web)