The Polybius square: letters as numbers and fire signals

In the tenth book of his Histories, Polybius of Megalopolis, the Greek historian who lived for many years in Rome in the second century BC, devotes a few pages to a practical problem: how to transmit any message over a distance, and not merely a signal agreed in advance. The solution he describes, based on torches and an orderly division of the alphabet, is the origin of what we now call the Polybius square.

The problem of fire signals

The Greeks had long used fires to announce expected events, such as the arrival of a fleet. The limitation, Polybius observes, was obvious: only news settled beforehand could be communicated, while war produces the unforeseen. Aeneas Tacticus, in the fourth century BC, had already proposed a system of synchronised water vessels with messages written on a rod, but that too allowed only a closed list of phrases.

Polybius credits two men, Cleoxenus and Democleitus, with the idea of a more flexible method, which he says he himself improved. The Greek alphabet of twenty-four letters was divided into five groups, each written on a tablet. The sender first raised on the left a number of torches equal to the group, and then on the right a number equal to the letter's position within the group.

From torches to the grid

The system required sighting instruments and good discipline, and we do not know how much it was actually used. Its importance, however, is conceptual: each letter is turned into a pair of numbers. Arranged in a grid of five rows and five columns, the letters acquire coordinates, like the squares of a chessboard. With the Latin alphabet two letters, such as I and J, are usually merged to arrive at twenty-five cells.

Originally the square was not a cipher, because anyone who knew the arrangement could read the signals. It becomes a tool of secrecy when the letters are placed in the grid in an order other than alphabetical, set for instance by a keyword, or when the resulting coordinates are themselves shuffled.

Prisons, revolutionaries and trenches

The Polybius grid had a long life as a knock code. In nineteenth-century Russian prisons inmates used it, according to many accounts, by tapping on cell walls, first for the row and then for the column. In the twentieth century the same principle was adopted by American prisoners of war in Vietnam, who called it the tap code. The grid also gave rise to the so-called Nihilist cipher, attributed to Russian revolutionaries.

The most famous military version was introduced by the German army in the spring of 1918, first in the form ADFGX and then, with the addition of V, as ADFGVX. The letters were placed in a grid whose rows and columns were labelled by those characters, chosen because they were easy to tell apart in Morse code; the resulting pairs then underwent a transposition. The French cryptanalyst Georges Painvin managed to decipher crucial messages between April and June 1918.

From Greek torches to knocks on a cell wall, the Polybius square shows how a simple idea of ordering can cross the centuries. Turning letters into numbers is, after all, the gesture underlying all modern communication, from the telegraph to computers, where every character is indeed represented by a number.