Английская Википедия:Frequency analysis
Шаблон:Short description Шаблон:Other uses
In cryptanalysis, frequency analysis (also known as counting letters) is the study of the frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers.
Frequency analysis is based on the fact that, in any given stretch of written language, certain letters and combinations of letters occur with varying frequencies. Moreover, there is a characteristic distribution of letters that is roughly the same for almost all samples of that language. For instance, given a section of English language, Шаблон:Samp, Шаблон:Samp, Шаблон:Samp and Шаблон:Samp are the most common, while Шаблон:Samp, Шаблон:Samp, Шаблон:Samp and Шаблон:Samp are rare. Likewise, Шаблон:Samp, Шаблон:Samp, Шаблон:Samp, and Шаблон:Samp are the most common pairs of letters (termed bigrams or digraphs), and Шаблон:Samp, Шаблон:Samp, Шаблон:Samp, and Шаблон:Samp are the most common repeats.[1] The nonsense phrase "ETAOIN SHRDLU" represents the 12 most frequent letters in typical English language text.
In some ciphers, such properties of the natural language plaintext are preserved in the ciphertext, and these patterns have the potential to be exploited in a ciphertext-only attack.
Frequency analysis for simple substitution ciphers
In a simple substitution cipher, each letter of the plaintext is replaced with another, and any particular letter in the plaintext will always be transformed into the same letter in the ciphertext. For instance, if all occurrences of the letter Шаблон:Samp turn into the letter Шаблон:Samp, a ciphertext message containing numerous instances of the letter Шаблон:Samp would suggest to a cryptanalyst that Шаблон:Samp represents Шаблон:Samp.
The basic use of frequency analysis is to first count the frequency of ciphertext letters and then associate guessed plaintext letters with them. More Шаблон:Samps in the ciphertext than anything else suggests that Шаблон:Samp corresponds to Шаблон:Samp in the plaintext, but this is not certain; Шаблон:Samp and Шаблон:Samp are also very common in English, so Шаблон:Samp might be either of them also. It is unlikely to be a plaintext Шаблон:Samp or Шаблон:Samp which are less common. Thus the cryptanalyst may need to try several combinations of mappings between ciphertext and plaintext letters.
More complex use of statistics can be conceived, such as considering counts of pairs of letters (bigrams), triplets (trigrams), and so on. This is done to provide more information to the cryptanalyst, for instance, Шаблон:Samp and Шаблон:Samp nearly always occur together in that order in English, even though Шаблон:Samp itself is rare.
An example
Suppose Eve has intercepted the cryptogram below, and it is known to be encrypted using a simple substitution cipher as follows:
Шаблон:Not a typo
For this example, uppercase letters are used to denote ciphertext, lowercase letters are used to denote plaintext (or guesses at such), and Шаблон:Samp~Шаблон:Samp is used to express a guess that ciphertext letter Шаблон:Samp represents the plaintext letter Шаблон:Samp.
Eve could use frequency analysis to help solve the message along the following lines: counts of the letters in the cryptogram show that Шаблон:Samp is the most common single letter,[2] Шаблон:Samp most common bigram, and Шаблон:Samp is the most common trigram. Шаблон:Samp is the most common letter in the English language, Шаблон:Samp is the most common bigram, and Шаблон:Samp is the most common trigram. This strongly suggests that Шаблон:Samp~Шаблон:Samp, Шаблон:Samp~Шаблон:Samp and Шаблон:Samp~Шаблон:Samp. The second most common letter in the cryptogram is Шаблон:Samp; since the first and second most frequent letters in the English language, Шаблон:Samp and Шаблон:Samp are accounted for, Eve guesses that Шаблон:Samp~Шаблон:Samp, the third most frequent letter. Tentatively making these assumptions, the following partial decrypted message is obtained.
Шаблон:Not a typo
Using these initial guesses, Eve can spot patterns that confirm her choices, such as "Шаблон:Samp". Moreover, other patterns suggest further guesses. "Шаблон:Samp" might be "Шаблон:Samp", which would mean Шаблон:Samp~Шаблон:Samp. Similarly "Шаблон:Samp" could be guessed as "Шаблон:Samp", yielding Шаблон:Samp~Шаблон:Samp and Шаблон:Samp~Шаблон:Samp. Furthermore, "Шаблон:Samp" might be "Шаблон:Samp", giving Шаблон:Samp~Шаблон:Samp. Filling in these guesses, Eve gets:
Шаблон:Not a typo
In turn, these guesses suggest still others (for example, "Шаблон:Samp" could be "Шаблон:Samp", implying Шаблон:Samp~Шаблон:Samp) and so on, and it is relatively straightforward to deduce the rest of the letters, eventually yielding the plaintext.
Шаблон:Not a typo
At this point, it would be a good idea for Eve to insert spaces and punctuation:
Hereupon Legrand arose, with a grave and stately air, and brought me the beetle from a glass case in which it was enclosed. It was a beautiful scarabaeus, and, at that time, unknown to naturalists—of course a great prize in a scientific point of view. There were two round black spots near one extremity of the back, and a long one near the other. The scales were exceedingly hard and glossy, with all the appearance of burnished gold. The weight of the insect was very remarkable, and, taking all things into consideration, I could hardly blame Jupiter for his opinion respecting it.
In this example from The Gold-Bug, Eve's guesses were all correct. This would not always be the case, however; the variation in statistics for individual plaintexts can mean that initial guesses are incorrect. It may be necessary to backtrack incorrect guesses or to analyze the available statistics in much more depth than the somewhat simplified justifications given in the above example.
It is also possible that the plaintext does not exhibit the expected distribution of letter frequencies. Shorter messages are likely to show more variation. It is also possible to construct artificially skewed texts. For example, entire novels have been written that omit the letter "Шаблон:Samp" altogether — a form of literature known as a lipogram.
History and usage
The first known recorded explanation of frequency analysis (indeed, of any kind of cryptanalysis) was given in the 9th century by Al-Kindi, an Arab polymath, in A Manuscript on Deciphering Cryptographic Messages.[3] It has been suggested that close textual study of the Qur'an first brought to light that Arabic has a characteristic letter frequency.[4] Its use spread, and similar systems were widely used in European states by the time of the Renaissance. By 1474, Cicco Simonetta had written a manual on deciphering encryptions of Latin and Italian text.[5]
Several schemes were invented by cryptographers to defeat this weakness in simple substitution encryptions. These included:
- Homophonic substitution: Use of homophones — several alternatives to the most common letters in otherwise monoalphabetic substitution ciphers. For example, for English, both X and Y ciphertext might mean plaintext E.
- Polyalphabetic substitution, that is, the use of several alphabets — chosen in assorted, more or less devious, ways (Leone Alberti seems to have been the first to propose this); and
- Polygraphic substitution, schemes where pairs or triplets of plaintext letters are treated as units for substitution, rather than single letters, for example, the Playfair cipher invented by Charles Wheatstone in the mid-19th century.
A disadvantage of all these attempts to defeat frequency counting attacks is that it increases complication of both enciphering and deciphering, leading to mistakes. Famously, a British Foreign Secretary is said to have rejected the Playfair cipher because, even if school boys could cope successfully as Wheatstone and Playfair had shown, "our attachés could never learn it!".
The rotor machines of the first half of the 20th century (for example, the Enigma machine) were essentially immune to straightforward frequency analysis. However, other kinds of analysis ("attacks") successfully decoded messages from some of those machines.[6]
Frequency analysis requires only a basic understanding of the statistics of the plaintext language and some problem solving skills, and, if performed by hand, tolerance for extensive letter bookkeeping. During World War II (WWII), both the British and the Americans recruited codebreakers by placing crossword puzzles in major newspapers and running contests for who could solve them the fastest. Several of the ciphers used by the Axis powers were breakable using frequency analysis, for example, some of the consular ciphers used by the Japanese. Mechanical methods of letter counting and statistical analysis (generally IBM card type machinery) were first used in World War II, possibly by the US Army's SIS. Today, the hard work of letter counting and analysis has been replaced by computer software, which can carry out such analysis in seconds. With modern computing power, classical ciphers are unlikely to provide any real protection for confidential data.
Frequency analysis in fiction
Frequency analysis has been described in fiction. Edgar Allan Poe's "The Gold-Bug", and Sir Arthur Conan Doyle's Sherlock Holmes tale "The Adventure of the Dancing Men" are examples of stories which describe the use of frequency analysis to attack simple substitution ciphers. The cipher in the Poe story is encrusted with several deception measures, but this is more a literary device than anything significant cryptographically.
See also
- ETAOIN SHRDLU
- Letter frequencies
- Arabic Letter Frequency
- Index of coincidence
- Topics in cryptography
- Zipf's law
- A Void, a novel by Georges Perec. The original French text is written without the letter e, as is the English translation. The Spanish version contains no a.
- Gadsby (novel), a novel by Ernest Vincent Wright. The novel is written as a lipogram, which does not include words that contain the letter E.
Further reading
- Helen Fouché Gaines, "Cryptanalysis", 1939, Dover. Шаблон:ISBN
- Abraham Sinkov, "Elementary Cryptanalysis: A Mathematical Approach", The Mathematical Association of America, 1966. Шаблон:ISBN.
References
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Ibrahim A. Al-Kadi "The origins of cryptology: The Arab contributions", Cryptologia, 16(2) (April 1992) pp. 97–126.
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
External links
- Online frequency analysis tool
- Character and syllable frequencies of 41 languages and a portable tool to create frequency and syllable distributions
- Arabic letter frequency analysis
- Conditional probabilities for characters in English text
- Czech letter/bigram/trigram frequency
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