Two-square Cryptanalysis
Like most pre-modern era ciphers, the two-square cipher can be easily cracked if there is enough text. Obtaining the key is relatively straightforward if both plaintext and ciphertext are known. 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.
Cryptanalysis of two-square almost always revolves around the transparency weakness. Depending on whether vertical or horizontal two-square was used, either the ciphertext or the reverse of the ciphertext should show a significant number of plaintext fragments. In a large enough ciphertext sample, there are likely to be several transparent digraphs in a row, revealing possible word fragments. 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.
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