The Two-square cipher, also called double Playfair, is a manual symmetric encryption technique.
This adds significant strength to the encryption when compared with monographic substitution ciphers, which operate on single characters.
The frequency analysis of digraphs is possible, but considerably more difficult, and it generally requires a much larger ciphertext in order to be useful.
Félix Delastelle described the cipher in his 1901 book Traité élémentaire de cryptographie under the name damiers bigrammatiques réduits (reduced digraphic checkerboard), with both horizontal and vertical types.
[3] Friedman's co-author on Military Cryptanalytics, Lambros D. Callimahos described the cipher in Collier's Encyclopedia in the Cryptography article.
The key can be written in the top rows of the table, from left to right, or in some other pattern, such as a spiral beginning in the upper-left-hand corner and ending in the center.
When only the ciphertext is known, brute force cryptanalysis of the cipher involves searching through the key space for matches between the frequency of occurrence of digraphs (pairs of letters) and the known frequency of occurrence of digraphs in the assumed language of the original message.
From these word fragments the analyst can generate candidate plaintext strings and work backwards to the keyword.
A good tutorial on reconstructing the key for a two-square cipher can be found in chapter 7, "Solution to Polygraphic Substitution Systems," of Field Manual 34-40-2, produced by the United States Army.