In classical cryptography

Cryptography

Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

, a permutation cipher is a transposition cipher

Transposition cipher

In cryptography, a transposition cipher is a method of encryption by which the positions held by units of plaintext are shifted according to a regular system, so that the ciphertext constitutes a permutation of the plaintext. That is, the order of the units is changed...

 in which the key is a permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

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To apply a cipher, a random permutation of size e is generated (the larger the value of e the more secure the cipher). The plaintext

Plaintext

In cryptography, plaintext is information a sender wishes to transmit to a receiver. Cleartext is often used as a synonym. Before the computer era, plaintext most commonly meant message text in the language of the communicating parties....

 is then broken into segments of size e and the letters within that segment are permuted according to this key.

In theory, any transposition cipher can be viewed as a permutation cipher where e is equal to the length of the plaintext; this is too cumbersome a generalisation to use in actual practice, however.

Identifying the cipher

Because the cipher doesn't change any of the characters, the ciphertext

Ciphertext

In cryptography, ciphertext is the result of encryption performed on plaintext using an algorithm, called a cipher. Ciphertext is also known as encrypted or encoded information because it contains a form of the original plaintext that is unreadable by a human or computer without the proper cipher...

 will have exactly the same letter frequencies as the underlying plaintext. This means that the cipher can in many cases be identified as a transposition by the close similarity of its letter statistics with the letter frequencies of the underlying language.

Breaking the cipher

Because the cipher operates on blocks of size e, the plaintext and the ciphertext have to have a length which is some multiple of e. This causes two weaknesses in the system: first, the plaintext may have to be padded (if the padding is identifiable then part of the key is revealed) and second, information relating to the length of the key is revealed by the length of the ciphertext. To see this, note that if the ciphertext is of length i then e must be one of the divisor

Divisor

In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

s of i. With the different possible key sizes different possible permutations are tried to find the permutation which results in the highest number of frequent bigrams

Bigram

Bigrams or digrams are groups of two written letters, two syllables, or two words, and are very commonly used as the basis for simple statistical analysis of text. They are used in one of the most successful language models for speech recognition...

 and trigrams

Trigram

Trigrams are a special case of the N-gram, where N is 3. They are often used in natural language processing for doing statistical analysis of texts.-Frequency:The 16 most common trigrams in English are:-Examples:...

 as found in the underlying language of the plaintext. Trying to find this permutation is essentially the same problem encountered when analysing a columnar transposition cipher: multiple anagramming

Anagram

An anagram is a type of word play, the result of rearranging the letters of a word or phrase to produce a new word or phrase, using all the original letters exactly once; e.g., orchestra = carthorse, A decimal point = I'm a dot in place, Tom Marvolo Riddle = I am Lord Voldemort. Someone who...

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