Rotor Machines

Rotor Machines

1. What a Rotor Machine Does

Rotor machines (Wiry, Enigma 1920s–40s) mechanized polyalphabetic substitution: each keystroke advances rotors, so the same plaintext letter maps differently every position — key period of $26^3 = 17{,}576$ (3 rotors) instead of Vigenère's short key. Passing current through wired rotors implements a monoalphabetic map that changes per keystroke:

Keyboard → Plugboard → Rotor R → Rotor M → Rotor L → Reflector
    → back through rotors → Plugboard → Lampboard

Current enters at plaintext letter, exits at ciphertext letter; rotors step like an odometer (right rotor every key, middle every 26, left every $26^2$).

2. Enigma Components (Exam Diagram Labels)

Part	Function	Key contribution
Rotors (Walzen) I–V, choice + order	Each a wired permutation ($26!$) + ring setting (Ringstellung)	Rotor choice/order: $5 \times 4 \times 3 = 60$; positions $26^3$; rings $26^2$ effective
Reflector (Umkehrwalze) B/C	Fixed involution sending current back — makes encryption = decryption (reciprocal) but NO letter ever maps to itself (crib flaw!)	Thin/betatable variants
Plugboard (Steckerbrett)	10 patch cables swapping letter pairs before/after rotors	$\binom{26}{20} \times$ pairings $\approx 1.5 \times 10^{14}$ — biggest key chunk
Ring setting + start position (Grundstellung)	Offsets wiring vs stepping	Daily key sheet entries

Total key space $\approx 10^{16} \times 10^{14} \approx 10^{23}$ ($2^{76}$-ish) — brute force impossible in 1930s; broken by procedure + mathematics, not key size.

3. How Encryption Stepping Works (Worked Trace Logic)

Press A: right rotor advances; circuit A → plugboard (say A↔M, so M) → rotor R map → M map → L map → reflector → back reversed → plugboard → lamp (say D). Press A again: rotor positions differ → new circuit → different lamp (say Q). Same input, different output — polyalphabetic with huge period. Decrypt: reset rotors to same start, type ciphertext — reflector's reciprocity returns plaintext (operator never flips a switch).

Numerical habit: given rotor I wiring "EKMFLGDQVZNTOWYHXUSPAIBRCJ", position offsets shift the mapping: output $= W(p + pos) - pos \bmod 26$ per rotor — exams ask one-rotor traces, not full Enigma.

4. Cryptanalysis: Why Enigma Fell

1. Procedural flaws: daily keys + message indicators sent twice (early war) → Polish bomba chained indicator relations to recover rotors (Rejewski, permutational math).
2. Cribs + no-self-encryption: Turing's bombe tested rotor positions against guessed plaintext ("WETTERBERICHT"); reflector guarantees $P_i
   e C_i$, killing positions where crib letter equals ciphertext — massive pruning.
3. Operator errors (Cillies, Herivel tip): lazy ring/position choices shrank key space.
4. Capture + Hollerith: key sheets, rotors from U-boats; mechanized search at Bletchley scaled rejection.

Lesson (links Topics 5–6, Module 2): big key space ≠ security — protocol/procedure flaws and a single structural regularity (reflector involution) defeat the machine. Modern ciphers demand no exploitable structure (avalanche, differential resistance) plus sound key management.

Exam Points

• Labelled current-path diagram + odometer stepping — 5-mark diagram core.
• Key-space contributors (60 orders, $26^3$ positions, plugboard $\sim 10^{14}$) — 2–3 mark numerical.
• Reflector: reciprocal encryption + never-self flaw — favourite justification.
• Break chain: indicators → bomba; cribs + bombe; operator errors.