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in cryptography, a substitution cipher is a method of encrypting by which units of plaintext are replaced with ciphertext, according to a fixed system; the "units" may be single letters (the most common), pairs of letters, triplets of letters, mixtures of the above, and so forth. the receiver deciphers the text by performing the inverse substitution. substitution ciphers can be compared with transposition ciphers. in a transposition cipher, the units of the plaintext are rearranged in a different and usually quite complex order, but the units themselves are left unchanged. by contrast, in a substitution cipher, the units of the plaintext are retained in the same sequence in the ciphertext, but the units themselves are altered. there are a number of different types of substitution cipher. if the cipher operates on single letters, it is termed a simple substitution cipher; a cipher that operates on larger groups of letters is termed polygraphic. a monoalphabetic cipher uses fixed substitution over the entire message, whereas a polyalphabetic cipher uses a number of substitutions at different positions in the message, where a unit from the plaintext is mapped to one of several possibilities in the ciphertext and vice versa. although the traditional keyword method for creating a mixed substitution alphabet is simple, a serious disadvantage is that the last letters of the alphabet (which are mostly low frequency) tend to stay at the end. a stronger way of constructing a mixed alphabet is to perform a columnar transposition on the ordinary alphabet using the keyword, but this is not often done. although the number of possible keys is very large, this cipher is not very strong, and is easily broken. provided the message is of reasonable length (see below), the cryptanalyst can deduce the probable meaning of the most common symbols by analyzing the frequency distribution of the ciphertext. this allows formation of partial words, which can be tentatively filled in, progressively expanding the (partial) solution (see frequency analysis for a demonstration of this). in some cases, underlying words can also be determined from the pattern of their letters; for example, attract, osseous, and words with those two as the root are the only common english words with the pattern abbcadb. many people solve such ciphers for recreation, as with cryptogram puzzles in the newspaper. according to the unicity distance of english, 27.6 letters of ciphertext are required to crack a mixed alphabet simple substitution. in practice, typically about 50 letters are needed, although some messages can be broken with fewer if unusual patterns are found. in other cases, the plaintext can be contrived to have a nearly flat frequency distribution, and much longer plaintexts will then be required by the cryptanalyst. the flag is ##############################