This presentation demonstrates the historical and mathematical background to the brilliant work done by Polish and British cryptology experts before and during World War II. The solutions provided by Marian Rejewski, Alan Turing and their co-workers had a major impact on the outcome of the war.

1. The Math That Saved the World Brad Young brad@clearpoint.co.il A Mathematical and Historical Analysis of the Cryptographic Attacks on the Nazi Enigma Machine Marian Rejewski Alan Turing

2. Agenda Development of Enigma Machine – Why/How/What The Rejewski Crack The Turing Crack Historical Impact

3. WWI Cryptology First major war with radio + telegraph Very large volume of communications Hand-ciphers Playfair, ADFGVX etc. Bigraph substitution + transformation Encryption/Decryption Inefficient …Became bottleneck Cryptanalysis Difficult, time-consuming… But successful (mainly)

4. Zimmermann Telegram

5. Invention of Enigma Machine Arthur Scherbius Efficient! (oh, and also Secure, by the way) Business, Military versions Early 1920’s – very poor sales German economy in trouble

6. Oops Publishes history book Reveals the impact of crypto on WWI Now, the Germans want Enigma!

7. A B C D E F G H Reflector 3rd Rotor 2nd Rotor 1st Rotor Lightbulbs Keyboard Enigma Schematic

8. A B C D E F G H Reflector Lightbulbs 3rd Rotor 2nd Rotor 1st Rotor Keyboard Electric Circuit

9. A B Pressing ‘A’ on the keyboard… C D E F … lights the ‘B’ lightbulb G H NOTE: Because it is a electric circuit, no letter can map to itself. Minor detail combinatorically speaking, but very important for the Turing crack. Reflector Lightbulbs 3rd Rotor 2nd Rotor 1st Rotor Keyboard Electric Circuit

10. A B C D E After each letter, the first rotor shifts one step. So now, pressing ‘A’ lights a different lightbulb….’F’ F G H Reflector Lightbulbs 3rd Rotor 2nd Rotor 1st Rotor Keyboard Rotor Shift

11. A B C D E F Sits between keyboard and rotors. Each plug cable swaps signal between two letters. 6 cables connect 12 letters. 14 other letters are not plugged at all. G H Reflector Lightbulbs Plugboard 3rd Rotor 2nd Rotor 1st Rotor Keyboard Plugboard

12. Plugboard

13. Keysize A B Rotor Order Rotor Setting Plugboard Wiring I – III - II VYJ A/G, D/Q, J/Z,L/S, M/V, N/T 3! = 6 263 =17,576 C(26,2) x C(24,2) x C(22,2) x C(20,2) x C(18,2) x C(16,2) x 1/6! (26!)3 x C(26,2)…C(2,2)x1/13! C ≈ 105 D E F ≈ 1011 ≈ 1092 G H Total Key Size ≈ 10108 Variable Key Size ≈ 1016

14. German Use of Enigma

16. Agenda Development of Enigma Machine – Why/How/What The Rejewski Crack The Turing Crack Historical Impact

17. Biuro Szyfrów 1918 – Polish Independence 1919 – Creation (and success) of Cipher Bureau 1926 – Germany goes dark as Enigma is adopted 1930 – Bring in the mathematicians (?!?) Marian Rejewski Jerzy Różycki Henryk Zygalski

18. The Rejewski Crack Intuition,Espionage,Engineering Understand how Enigma works Reverse-engineer the wiring Be able to crack the key each day Permutational Mathematics

19. The Math of Permutation Cycles P = P-1 =

20. Cycle Notation P = P = (AECH)(BFD)(G) = (BFD)(G) (AECH) = (FDB)(G)(CHAE) P-1 = (HCEA)(DFB)(G) Benefits of cycle notation: Concise Easier to take inverse (These are benefits of efficiency)

21. Cycle Structure = (AECH)(BFD)(G) P = 4 3 1 = (AFC)(BG)(D)(EH) Q = 3 2 1 2 Benefits of cycle notation: Concise Easier to take inverse Gives more info – Cycle Structure (This is a benefit of value-add information)

22. Composition P = = (AECH)(BFD)(G) Q = = (AFC)(BG)(D)(EH) Q ◦ P = Q(P()) = (AHFDGBCE) Q ◦ P ≠ P ◦ Q - NOT Commutative Q ◦ ( P ◦ R ) = ( Q ◦ P ) ◦ R - Associative

23. Identity = (A)(B)(C)(D)(E)(F)(G)(H) I = P ◦ I = I ◦ P = P P ◦ P -1 = I I ◦ I = I i.e. I = I -1 (ab) ≠ I , but (ab) ◦ (ab) = (a)(b) i.e. (ab) = (ab)-1

24. Conjugation Conjugation of Q by P is defined as P ◦ Q ◦ P-1 P = (AECH)(BFD)(G) P-1 = (HCEA)(DFB)(G) Q = (AFC)(BG)(D)(EH) 1-2-2-3 1-2-2-3 This is not a coincidence! This is not a coincidence! P ◦ Q ◦ P-1 = (AC)(B)(DHE)(FG)

25. Theorem: Cycle structure is invariant under conjugation Proof: Suppose Q: ij, that is Q(i) = j. Consider P ◦ Q ◦ P-1 (P(i)). P ◦ Q ◦ P-1 (P(i)) = P ◦ Q ◦ (P-1 ◦ P)(i) = P ◦ Q(i) = P(j) i.e. P ◦ Q ◦ P-1: P(i)P(j) Therefore… If Q has k-cycle (i1, i2 … ik) then P ◦ Q ◦ P-1 has k-cycle (P(i1), P(i2)…P(ik)) QED

26. Using Permuation Cycles on Enigma A B Suppose we intercept a message: BOLJRVSQIGPQTMNWJRAKOBYTKMTTGBBRQUPWLHSOLNFEQTHJOVX Plaintext: abcabcCiphertext: BOLJRV Define En as the permutation that occurs when Enigma machine is in state n. So, in the first state, aB. In the fourth state, aJ E1 = (aB …E4 = (aJ … Now…Recall the effect of the Reflector, which creates 2-letter circuits So, if aB, then Ba. So the cycle is closed. E1 = (aB) …E4 = (aJ) … So, we can now compute E4 ◦ E1 = (BJ … C These are the variablesa,b,c, not the actual letters D E F G H

27. Using Permuation Cycles on Enigma If we have many intercepts from the same day, then they were produced with the same day settings. So we can calculate the entire compositions… E4 ◦ E1 = (BJUMPWTCFE)(ARDNHSLYZK)(G)(I)(O)(Q)(X)(V)E5 ◦ E2 = (ORJCLVHGXKF)(AUYMPZQNDWB)(ES)(IT)E6 ◦ E3 = (BWOIKTZHXB)(EPQJYLVGN)(ARCU)(DSMF) Good news: abc variables have been eliminated! We’ve found a unique identifier! Bad news: It is one of 10,000,000,000,000,000 possibilities

28. Explore the nature of En A B En = P ◦ Rn ◦ P where P is the plugboard permutation and Rn is rotor permutation when in state n E4 ◦ E1 = P ◦ R4 ◦ P ◦ P ◦ R1 ◦ P Now, recall the plugboard… P = (ab)(cd)(ef)(gh)(ij)(kl)(m)(n)(o)(p)(q)(r)(s)(t)(u)(v)(w)(x)(y)(z) All 2-cycles and 1-cycles, therefore P = P-1 ! E4 ◦ E1 = P ◦ R4 ◦ P ◦ P ◦ R1 ◦ P = P ◦ R4 ◦ P ◦ P-1 ◦ R1 ◦ P = P ◦ R4 ◦ (P ◦ P-1 ) ◦ R1 ◦ P = P ◦ R4 ◦ R1 ◦ P = P ◦ (R4 ◦ R1 ) ◦ P = P ◦ (R4 ◦ R1 ) ◦ P-1 C P R D E F G H Conjugation:Cycle structure of E4 ◦ E1 is same as cycle structure of R4 ◦ R1 and is not affected at all by the plugboard! E4 ◦ E1 = (BJUMPWTCFE)(ARDNHSLYZK)(G)(I)(O)(Q)(X)(V)E5 ◦ E2 = (AUYMPZQNDWB)(CLVHGXKFORJ)(ES)(IT)E6 ◦ E3 = (BWOIKTZHXB)(EPQJYLVGN)(ARCU)(DSMF) 1-1-1-1-1-1-10-10 ; 2-2-11-11 ; 4-4-9-9 Remember: Keysize(R) ≈ 105 Keysize(P) ≈ 1011

29. Now, where are we? Figuring out En is problem of size 1016 Now, we have Rn, a smaller problem: 105 Just barely small enough to attack brute force

30. Building the Rejewski Dictionary RO RS E4 ◦ E1 E5 ◦ E2 E6 ◦ E3 1 2 3 AAA 13-13 1-1-12-12 1-1-12-12 1 2 3 BAA 1-1-12-12 1-1-12-12 2-2-11-11 1 2 3 CAA 1-1-12-12; 2-2-11-11 1-1-12-12 1 2 3 DAA 2-2-11-11 1-1-12-12 13-13 1 2 3 EAA 1-1-12-12 13-13 13-13 1 2 3 FAA 13-13 13-13 1-1-2-2-3-3-3-3-4-4 1 2 3 GAA 13-13 1-1-2-2-3-3-3-3-4-4 2-2-5-5-6-6 1 2 3 HAA 1-1-2-2-3-3-3-3-4-4 2-2-5-5-6-6 13-13 1 2 3 IAA 2-2-5-5-6-6 13-13 4-4-9-9 1 2 3 JAA 13-13 4-4-9-9 1-1-5-5-7-7 1 2 3 KAA 4-4-9-9 1-1-5-5-7-7 13-13 1 2 3 LAA 1-1-5-5-7-7 13-13 1-1-2-2-10-10 1 2 3 MAA 13-13 1-1-2-2-10-10 1-1-1-1-11-11 . . . . . . . . . . . . . . . … 2-2-11-11; 1-1-1-1-1-1-1-1-4-4-5-5; 1-1-12-12 KFE 213 2-2-11-11; 1-1-1-1-1-1-1-1-4-4-5-5; 2-2-5-5-6-6 ZTF 132 2-2-11-11; 1-1-1-1-1-1-1-1-4-4-5-5; 5-5-8-8 GIC 312 2-2-11-11; 1-1-1-1-1-1-1-1-9-9; 1-1-12-12 AHH 132 2-2-11-11; 1-1-1-1-1-1-1-1-9-9; 1-1-12-12 WLA 312 2-2-11-11; 1-1-1-1-1-1-1-1-9-9; 1-1-5-5-7-7 YKG 132 2-2-11-11; 1-1-1-1-1-1-1-1-9-9; 13-13 DXI 213 2-2-11-11; 1-1-1-1-1-1-1-1-9-9; 13-13 ESY 321 2-2-11-11; 1-1-1-1-1-1-1-1-9-9; 13-13 VHX 213 2-2-11-11; 1-1-1-1-1-1-1-1-9-9; 2-2-11-11 UNV 231 … 1 setting every 4 minutes, x 20 hours/day = 300 / day 105 / 300 ≈ 1 year to complete Good news; Solved the RO, RS! Bad news: 105 solved, 1011 not solved Cycle structure is not unique …even though 105 << (1012)3 ≈ 1012 But most have < 10

31. Recovering the Plugboard Plugboard is the biggest problem combinatorically But… It is trivial to solve E4 ◦ E1 = (BJUMPWTCFE)(ARDNHSLYZK)(G)(I)(O)(Q)(X)(V) R4 ◦ R1 = (MGWTREFBJU)(AKZCINLSHY)(P)(D)(O)(Q)(V)(X) (BJUMPWTCFE) (BJUMGWTREF) Plugboard settings: P/G , C/R , E/F , etc.

32. Paradox of Decreasing Benefit Keysize # Cables

33. Agenda Development of Enigma Machine – Why/How/What The Rejewski Crack The Turing Crack Historical Impact

34. 1939 – Brink of War Polish deliver Enigma replica and training to England and France Biuro Szyfrów is dismantled

35. Bletchley Park HQ of British Government Code and Cypher School (GCCS)

36. New Challenges Combinatoric More rotors to choose from Increase # of plugs Ring settings Procedural Eliminate Message Key repetition Navy / Air Force / Army mods Keysize now 1023

37. Turing’s Solution Known-Plaintext attack Heil Hitler Wetterbericht Seeding values Plaintext Crib:Ciphertext: Try to place the crib without letter any letter mapping to itself WETTERBERICHT WETTERBERICHT WETTERBERICHT WETTERBERICHT WETTERBERICHT EXLMBTWZXBITWZCIQ P(false hit) = (25/26)length of crib

38. Finding Cycles WETTERBERICHT EXLMBTWZXBITW E1: WE E5: EB E7: BW

39. J Q F b E E1 E1: WE E5: EB E7: BW a W J Q J B b E E5 c J Q L B E7 c a W

40. J Q F b E1 a J Q J b E5 c J Q L E7 c a

41. M V C b E1 a M Z C b E5 c M B D E7 c a

42. M V C b a E1 M Z C b E5 c M B D E7 P(false hit) = (1/26)length of cycle-1 a c

43. Turing’s Bombe NOT a computer Multi-Enigma Wiring 120 rpm  max 6 hrs to solve ~70% of days cracked Accurate crib? Location of crib in message? Find cycle in message? Not too many false hits? Crib seeding Fake missions – Get spotted 18’26”N, 72’49”E = einachtzweisechsnordensiebenzweivierneunosten Reimann zeta zeros

44. Agenda Development of Enigma Machine – Why/How/What The Rejewski Crack The Turing Crack Historical Impact

45. 6 : 60,000,000 :: 8 : ?

46. Secrecy Bletchley Park is gutted Enigma machines captured (and distributed!) Top Secret status until 1973!

48. 1939 – France

49. French cipher bureau

50. 1940 – Algeria

51. 1940 – Back to France

52. Rozycki dies in transit

53. Underground cryptography

54. 1942 – Spain

55. Betrayed mid-crossing

56. Arrested + Jailed

57. 1942 – Portugal, Gibraltar

58. 1942 – England

59. No security clearance (Vichy France)

60. Polish Army – hand ciphers

61. 1945 – Poland

62. 1950 – Cable salesman

63. Secret Service meddling

64. 1955 – Bookkeeper

65. Until retirement

66. 1973 – Finally learns about ULTRA

68. Colossus Computer Cracks Lorenz cipher High-level German communications History of Computers Z3 Colossus ENIAC Mark I

69. NSA

70. Addenda, Errata, Anecdotes Wiring analysis Hans Thilo-Schmidt TTTTTTTTTTTT Entry wheel order Why E1-E6, instead of E0-E5 ? Ring Settings and Rotor Stepping “Turing. Alan Turing.” Other WWII Cryptanalysis Disguising ULTRA intelligence Suggested Reading David Kahn – The Codebreakers Simon Singh – The Code Book