The Electromechanical Bombe and the Diagonal Board

Turning Enigma's Self-Reciprocal Flaw Into a Parallel Contradiction Engine

Every day at midnight during the Second World War, German military radio operators reset their portable Enigma cipher machines to a new daily key printed on water-soluble paper sheets. Inside a standard three-rotor Wehrmacht or Luftwaffe Enigma sat a keyboard, a front plugboard (Steckerbrett) that swapped up to ten pairs of letters via patch cords, three rotating scrambler wheels chosen from a box of five, a reversing reflector (Umkehrwalze), and twenty-six glow lamps. Pressing a key sent current through the plugboard, forward through all three rotors, around the reflector, backward through the rotors, and out through the plugboard to light a lamp. Across the sixty wheel orders, $17{,}576$ rotor start positions, ring settings, and $1.5 \times 10^{14}$ plugboard arrangements, an eavesdropper faced roughly $1.58 \times 10^{20}$ daily configurations.

Between 1932 and 1938, Polish Cipher Bureau mathematician Marian Rejewski and his colleagues Jerzy Różycki and Henryk Zygalski broke early German Enigma traffic by exploiting a procedural repetition. German operators encrypted a three-letter message indicator twice at the start of every message (ABC ABC). Rejewski designed an electromechanical rotor-stepping machine called the Bomba that hunted for indicator cycles. When Polish intelligence handed their Bomba blueprints and replica Enigma machines to British and French cryptanalysts at Pyry near Warsaw in July 1939, Alan Turing and Gordon Welchman at Bletchley Park saw a fatal vulnerability. Once the Germans stopped doubling the three-letter message key—which they did in May 1940—Rejewski's Bomba would stop working overnight.

Working in Hut 8 in the autumn of 1939, Turing designed a new machine—the British Bombe—that abandoned indicator doubling in favor of known-plaintext cribs. German radiomen routinely began morning dispatches with predictable stereotypes such as WETTERVORHERSAGE (weather forecast). Because the Enigma reflector sent current back through the rotors on a different wire, no letter could ever encrypt to itself ($E(x) \neq x$). A cryptanalyst slid a candidate crib along an intercepted ciphertext until zero letters matched vertically. From that alignment, the analyst drew a menu graphing which plaintext letters encrypted to which ciphertext letters at offsets $1, 2, \dots, k$.

The core obstacle was the plugboard. Even if the three rotor wheels were known, $1.5 \times 10^{14}$ plugboard combinations masked the letters entering and leaving the drums. Turing bypassed the plugboard by wiring closed electrical loops in the crib menu. Suppose position 1 links A to E, position 5 links E to R, and position 12 links R back to A. One connects three double-ended Enigma scramblers in series set to offsets 1, 5, and 12. Any unknown plugboard partner $S(\text{A}) = \alpha$ entering the first scrambler must emerge from the third scrambler as the exact same letter $\alpha$ when the rotor positions are right.

Instead of testing all twenty-six possible plug partners $\alpha$ one by one, Turing applied voltage to a single wire on the twenty-six-wire A bundle. Electricity rippled around the loop at the speed of light. At an incorrect rotor orientation, the hypothesis $S(\text{A}) = \alpha$ propagated contradictions that energized all twenty-six wires in the register. The machine ignored that flood and spun on. Only at a logically consistent rotor orientation did either a single wire stay energized (meaning the guess $\alpha$ was right) or twenty-five wires light up while one stayed dark (meaning the single unlit wire was the true plugboard partner).

In early 1940, Gordon Welchman realized that Enigma's plugboard was reciprocally symmetric: if A was plugged to G, then G was plugged to A ($S(x) = y \iff S(y) = x$). Welchman added a $26 \times 26$ grid of cross-soldered contacts called the diagonal board, wiring terminal $y$ of bundle $x$ to terminal $x$ of bundle $y$. Current now flowed across branching trees as well as closed loops, eliminating almost all false stops. Engineered by Harold Keen at the British Tabulating Machine Company in Letchworth, the first one-ton Bombe (Victory) arrived at Bletchley Park in March 1940, followed by the diagonal-board Agnus Dei Bombe in August 1940. By the end of the Second World War, more than two hundred British Bombes spinning in Letchworth and outstations around Bletchley Park were testing thousands of Enigma rotor orders every day and turning electrical short circuits into naval and air intelligence.