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Caesar cipher
Caesar cipher
Caesar cipher
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The action of a Caesar cipher is to replace each plaintext letter with a different one a fixed number of places down the alphabet. The cipher illustrated here uses a left shift of 3, so that (for example) each occurrence of E in the plaintext becomes B in the ciphertext.

In cryptography, a Caesar cipher, also known as Caesar's cipher, the shift cipher, Caesar's code, or Caesar shift, is one of the simplest and most widely known encryption techniques. It is a type of substitution cipher in which each letter in the plaintext is replaced by a letter some fixed number of positions down the alphabet. For example, with a left shift of 3, D would be replaced by A, E would become B, and so on.[1] The method is named after Julius Caesar, who used it in his private correspondence.

The encryption step performed by a Caesar cipher is often incorporated as part of more complex schemes, such as the Vigenère cipher, and still has modern application in the ROT13 system. As with all single-alphabet substitution ciphers, the Caesar cipher is easily broken and in modern practice offers essentially no communications security.

Example

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The transformation can be represented by aligning two alphabets; the cipher is the plain alphabet rotated left or right by some number of positions. For instance, here is a Caesar cipher using a left rotation of three places, equivalent to a right shift of 23 (the shift parameter is used as the key):

Plain ABCDEFGHIJKLMNOPQRSTUVWXYZ
Cipher XYZABCDEFGHIJKLMNOPQRSTUVW

When encrypting, a person looks up each letter of the message in the "plain" line and writes down the corresponding letter in the "cipher" line. 

Plaintext:  THE QUICK BROWN FOX JUMPS OVER THE LAZY DOG
Ciphertext: QEB NRFZH YOLTK CLU GRJMP LSBO QEB IXWV ALD

Deciphering is done in reverse, with a right shift of 3.

The encryption can also be represented using modular arithmetic by first transforming the letters into numbers, according to the scheme, A → 0, B → 1, ..., Z → 25.[2] Encryption of a letter x by a shift n can be described mathematically as,[3][4]

Decryption is performed similarly,

(Here, "mod" refers to the modulo operation. The value x is in the range 0 to 25, but if x + n or x − n are not in this range then 26 should be added or subtracted.)

The replacement remains the same throughout the message, so the cipher is classed as a type of monoalphabetic substitution, as opposed to polyalphabetic substitution.

History and usage

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See also: History of cryptography
The Caesar cipher is named for Julius Caesar, who used an alphabet where decrypting would shift three letters to the left.

The Caesar cipher is named after Julius Caesar, who, according to Suetonius, used it with a shift of three (A becoming D when encrypting, and D becoming A when decrypting) to protect messages of military significance.[5] While Caesar's was the first recorded use of this scheme, other substitution ciphers are known to have been used earlier.[6][7]

"If he had anything confidential to say, he wrote it in cipher, that is, by so changing the order of the letters of the alphabet, that not a word could be made out. If anyone wishes to decipher these, and get at their meaning, he must substitute the fourth letter of the alphabet, namely D, for A, and so with the others."

— Suetonius, Life of Julius Caesar 56

His nephew, Augustus, also used the cipher, but with a right shift of one, and it did not wrap around to the beginning of the alphabet:

"Whenever he wrote in cipher, he wrote B for A, C for B, and the rest of the letters on the same principle, using AA for Z."

— Suetonius, Life of Augustus 88

Evidence exists that Julius Caesar also used more complicated systems,[8] and one writer, Aulus Gellius, refers to a (now lost) treatise on his ciphers:

"There is even a rather ingeniously written treatise by the grammarian Probus concerning the secret meaning of letters in the composition of Caesar's epistles."

— Aulus Gellius, Attic Nights 17.9.1–5

It is unknown how effective the Caesar cipher was at the time; there is no record at that time of any techniques for the solution of simple substitution ciphers. The earliest surviving records date to the 9th-century works of Al-Kindi in the Arab world with the discovery of frequency analysis.[9]

A piece of text encrypted in a Hebrew version of the Caesar cipher not to be confused with Atbash, is sometimes found on the back of Jewish mezuzah scrolls. When each letter is replaced with the letter before it in the Hebrew alphabet the text translates as "YHWH, our God, YHWH", a quotation from the main part of the scroll.[10][11]

In the 19th century, the personal advertisements section in newspapers would sometimes be used to exchange messages encrypted using simple cipher schemes. David Kahn (1967) describes instances of lovers engaging in secret communications enciphered using the Caesar cipher in The Times.[12] Even as late as 1915, the Caesar cipher was in use: the Russian army employed it as a replacement for more complicated ciphers which had proved to be too difficult for their troops to master; German and Austrian cryptanalysts had little difficulty in decrypting their messages.[13]

Caesar cipher can be constructed into a disk with outer rotating wheel as plain text and the inner fixed wheel as cipher text. Both outer and inner plates should have alphabets in the same direction.
Caesar cipher translated to a disk has both outer and inner plates having alphabets in the same direction and not the reverse as seen in CipherDisk2000.

Caesar ciphers can be found today in children's toys such as secret decoder rings. A Caesar shift of thirteen is also performed in the ROT13 algorithm, a simple method of obfuscating text widely found on Usenet and used to obscure text (such as joke punchlines and story spoilers), but not seriously used as a method of encryption.[14]

The Vigenère cipher uses a Caesar cipher with a different shift at each position in the text; the value of the shift is defined using a repeating keyword.[15] If the keyword is as long as the message, is chosen at random, never becomes known to anyone else, and is never reused, this is the one-time pad cipher, proven unbreakable. However the problems involved in using a random key as long as the message make the one-time pad difficult to use in practice. Keywords shorter than the message (e.g., "Complete Victory" used by the Confederacy during the American Civil War), introduce a cyclic pattern that might be detected with a statistically advanced version of frequency analysis.[16]

In April 2006, fugitive Mafia boss Bernardo Provenzano was captured in Sicily partly because some of his messages, clumsily written in a variation of the Caesar cipher, were broken. Provenzano's cipher used numbers, so that "A" would be written as "4", "B" as "5", and so on.[17]

In 2011, Rajib Karim was convicted in the United Kingdom of "terrorism offences" after using the Caesar cipher to communicate with Bangladeshi Islamic activists discussing plots to blow up British Airways planes or disrupt their IT networks. Although the parties had access to far better encryption techniques (Karim himself used PGP for data storage on computer disks), they chose to use their own scheme (implemented in Microsoft Excel), rejecting a more sophisticated code program called Mujahedeen Secrets "because 'kaffirs', or non-believers, know about it, so it must be less secure".[18]

Breaking the cipher

[edit]
Decryption
shift 		Candidate plaintext
0 exxegoexsrgi
1 dwwdfndwrqfh
2 cvvcemcvqpeg
3 buubdlbupodf
4 attackatonce
5 zsszbjzsnmbd
6 yrryaiyrmlac
...
23 haahjrhavujl
24 gzzgiqgzutik
25 fyyfhpfytshj

The Caesar cipher can be easily broken even in a ciphertext-only scenario. Since there are only a limited number of possible shifts (25 in English), an attacker can mount a brute force attack by deciphering the message, or part of it, using each possible shift. The correct description will be the one which makes sense as English text.[19] An example is shown on the right for the ciphertext "exxegoexsrgi"; the candidate plaintext for shift four "attackatonce" is the only one which makes sense as English text. Another type of brute force attack is to write out the alphabet beneath each letter of the ciphertext, starting at that letter. Again the correct decryption is the one which makes sense as English text. This technique is sometimes known as "completing the plain component".[20][21]

The distribution of letters in a typical sample of English language text has a distinctive and predictable shape. A Caesar shift "rotates" this distribution, and it is possible to determine the shift by examining the resultant frequency graph.

Another approach is to match up the frequency distribution of the letters. By graphing the frequencies of letters in the ciphertext, and by knowing the expected distribution of those letters in the original language of the plaintext, a human can easily spot the value of the shift by looking at the displacement of particular features of the graph. This is known as frequency analysis. For example, in the English language the plaintext frequencies of the letters E, T, (usually most frequent), and Q, Z (typically least frequent) are particularly distinctive.[22] Computers can automate this process by assessing the similarity between the observed frequency distribution and the expected distribution. This can be achieved, for instance, through the utilization of the chi-squared statistic[23] or by minimizing the sum of squared errors between the observed and known language distributions.[24]

The unicity distance for the Caesar cipher is about 2, meaning that on average at least two characters of ciphertext are required to determine the key.[25] In rare cases more text may be needed. For example, the words "river" and "arena" can be converted to each other with a Caesar shift, which means they can produce the same ciphertext with different shifts. However, in practice the key can almost certainly be found with at least 6 characters of ciphertext.[26]

With the Caesar cipher, encrypting a text multiple times provides no additional security. This is because two encryptions of, say, shift A and shift B, will be equivalent to a single encryption with shift A + B. In mathematical terms, the set of encryption operations under each possible key forms a group under composition.[27]

See also

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Notes

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Bibliography

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Further reading

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Caesar cipher

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from Grokipedia
The Caesar cipher is a monoalphabetic in which each letter of the message is replaced by a letter a fixed number of positions further in the , typically wrapping around from Z to A, with the shift amount serving as the secret key. Named after the Roman general and statesman , who employed it around the 50s BCE to confidential military orders sent to his troops, the method is one of the earliest documented techniques. According to the Roman historian in his The Twelve Caesars (c. 121 CE), Caesar specifically used a shift of three positions: for instance, A became D, B became E, and so on, ensuring that "not a word could be made out" without applying the reverse substitution. This shift operates modulo 26 for the , formalized as letter = (plaintext position + key) mod 26 for , and plaintext = ( position - key) mod 26 for decryption, making it straightforward to implement but highly vulnerable to brute-force attacks since there are only 25 possible non-trivial keys. An example encryption of the plaintext "MEET ME AFTER THE " with a key of 3 yields "PHHW PH DIWHU WKH WRJD SDUWB," preserving word lengths and letter frequencies while obscuring meaning. Despite its simplicity, the Caesar cipher's security relies solely on the secrecy of the key, and it was first systematically broken in the 9th century CE by the Arab polymath through , which exploits the predictable distribution of letters in (e.g., being the most common in English). This vulnerability limited its practical use in antiquity, though variants appeared in later ciphers like the Vigenère square. In modern contexts, the Caesar cipher serves primarily as an educational tool for introducing cryptographic principles, and it inspired —a fixed shift of 13 positions used for mild in online text, such as rot13.com for reversible encoding without needing a shared key.

Fundamentals

Definition

The Caesar cipher is a monoalphabetic that replaces each letter in the with another letter a fixed number of positions further in the , using a consistent shift for the entire message. This method creates a direct mapping between the original and a shifted version of itself, preserving the relative order of letters while obscuring the original text. In contrast to general monoalphabetic substitution ciphers, which allow arbitrary rearrangements of the , the Caesar cipher restricts the transformation to a simple cyclic shift determined by a single parameter, making it a foundational example of symmetric key encryption. The key consists of the shift value k, an integer generally ranging from 1 to 25 in a 26-letter to exclude the trivial no-shift case, with the classical variant employing k=3. It is attributed to , who used such a shift for securing private messages. The cipher operates on standard alphabets like the Latin (A-Z), typically leaving non-alphabetic characters unchanged and treating the process as case-insensitive in its basic form, though modern adaptations may preserve case.

Mechanics

The Caesar cipher operates by systematically shifting the letters of the plaintext alphabet by a fixed number of positions, known as the key kk, where 0≤k<260 \leq k < 26. To formalize this, each letter in the plaintext is first mapped to its numerical position pp, with A (or a) assigned 0, B (or b) assigned 1, up to Z (or z) assigned 25. The corresponding ciphertext letter is then obtained via the encryption formula c=(p+k)mod  26c = (p + k) \mod 26, where the result cc determines the shifted position in the alphabet. Decryption reverses this process using the formula p=(c−k)mod  26p = (c - k) \mod 26, which shifts the ciphertext letters back by kk positions to recover the original positions. The operation mod  26\mod 26 ensures the shifting wraps around the cyclically: for instance, shifting Z (25) forward by 1 yields A (0), as (25+1)mod  26=0(25 + 1) \mod 26 = 0, preventing overflow beyond the 26-letter boundary. Non-alphabetic characters, such as spaces, , or numbers, are typically left unchanged during both and decryption to preserve the message's . Regarding , implementations often standardize the text to uppercase or lowercase for processing, though some preserve the original case by applying the shift separately to uppercase and lowercase alphabets. A classical example uses k=3k=3, shifting each letter forward by three positions.

Historical Context

Origins

The Caesar cipher is attributed to (100–44 BCE), who employed it to secure confidential communications during his time as a Roman general and statesman. According to the Roman historian in his biographical work De Vita Caesarum (The Lives of the Twelve Caesars), completed around 121 CE, Caesar wrote letters to figures like and his close associates using a substitution method to obscure sensitive content from potential interceptors. Suetonius notes that Caesar shifted each letter in the Latin alphabet by three positions, such that A became D, B became E, and so on, rendering the text unintelligible without the key. Other 2nd-century Roman authors, including and , also described this cipher in their works. This account by represents the earliest documented description of the , dating to the early CE, though Caesar's usage likely occurred during his military campaigns in the late , particularly the (58–50 BCE). While ancient civilizations, such as the with the Atbash substitution around 600 BCE, developed other forms of letter replacement, no concrete evidence exists for a systematic shift prior to the Roman era. The 's invention aligns with the needs of Roman expansion, where secure transmission of orders and intelligence was essential amid frequent interceptions by enemies. In the classical Roman context, the facilitated both dispatches and political correspondence, protecting strategic information during the turbulent final decades of the . emphasizes its application in private letters containing confidential matters, underscoring its role in maintaining among elites in an era of civil strife and . This early form of thus marked a foundational step in cryptographic practice, tailored to the Latin alphabet and the demands of Roman governance.

Usage

Following its initial adoption in ancient Roman military communications, the Caesar cipher saw renewed use in medieval and as a straightforward method for secret writing, particularly in diplomatic exchanges to safeguard confidential information from interception. By the late , European states employed substitution ciphers, including shift-based techniques similar to the Caesar method, for official correspondence amid rising concerns during conflicts and alliances. In , such enciphered dispatches were standard by 1411, drawing on classical Roman precedents documented by , with professional codebreakers like Giovanni Soro refining these systems for state in the early . The cipher's cultural prominence grew in the 19th century through literary works, most notably Edgar Allan Poe's 1843 "The Gold-Bug," which centered on a puzzle solved through , sparking widespread public interest in as an intellectual pursuit. Poe's , featuring William Legrand decoding a to uncover , exemplified the role of substitution ciphers in early and popularized code-solving as a recreational challenge. In the , a variant known as —employing a fixed shift of 13 positions—became prevalent on newsgroups starting in the early , primarily to obscure spoilers in discussions of films, books, and events, as well as potentially offensive humor, allowing voluntary decoding by interested readers. This self-inverse transformation, which decodes identically when applied twice, facilitated quick online encoding and remains supported by tools like rot13.com for casual text obfuscation in forums and emails. Outside secure communications, the Caesar cipher serves extensively in non-military domains, including puzzles, board games, and educational curricula designed to build foundational skills. It appears in activities like code-cracking challenges in science museums and classrooms, where participants shift letters to messages and learn about substitution patterns without needing advanced tools. Such applications emphasize conceptual understanding over protection, fostering problem-solving in subjects like and history. Although historically versatile, the Caesar cipher is seldom applied in practice for genuine , given its susceptibility to basic attacks like exhaustive key testing across only 25 possible shifts, rendering it ineffective against determined adversaries. It endures instead for trivial , such as hiding puzzle solutions or temporary text in low-stakes environments.

Practical Illustration

Encryption Example

To illustrate the encryption process of the Caesar cipher, consider the uppercase "THEQUICKBROWNFOX", the well-known phrase "THE QUICK BROWN FOX" (omitting spaces for clarity), encrypted with the classical shift of k=3 as used by . Each letter's position in the (A=0, B=1, ..., Z=25) is increased by 3 26 to determine the letter. The step-by-step transformation begins with T at position 19, yielding (19 + 3) mod 26 = 22, which corresponds to W; H at position 7 becomes (7 + 3) mod 26 = 10 or K; E at 4 becomes 7 or H; Q at 16 becomes 19 or T; U at 20 becomes 23 or X; I at 8 becomes 11 or L; C at 2 becomes 5 or F; K at 10 becomes 13 or N; B at 1 becomes 4 or E; R at 17 becomes 20 or U; O at 14 becomes 17 or R; W at 22 becomes 25 or Z; N at 13 becomes 16 or Q; F at 5 becomes 8 or I; the second O at 14 becomes 17 or R; and X at 23 becomes (23 + 3) mod 26 = 0 or A. The resulting is "WKHTXLFNEURZQIRA".

Plaintext alphabet: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Ciphertext alphabet: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

Plaintext alphabet: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Ciphertext alphabet: D E F G H I J K L M N O P Q R S T U V W X Y Z A B C

This diagram shows the uniform shift applied to the entire alphabet. In practice, non-letter characters like spaces and punctuation are preserved to maintain readability. For example, the plaintext "THE QUICK" encrypts to "WKH TXLFN".

Decryption Example

To decrypt a Caesar cipher, the recipient reverses the shift applied during encryption, assuming the key kk is known. For a shift of k=3k=3, each ciphertext letter is shifted backward by 3 positions in the alphabet, wrapping around from A to Z if necessary; numerically, this is computed as plaintext position P=(C−k)mod  26P = (C - k) \mod 26, where CC is the ciphertext letter's position (A=0, B=1, ..., Z=25). Consider the ciphertext "WKHTXLFNEURZQIRA", which was encrypted from the "THEQUICKBROWNFOX" using k=3k=3. Starting with the first letter, W (position 22) subtracts 3 to yield 19, corresponding to T. The second letter K (position 10) subtracts 3 to yield 7 (H). Continuing this process: H (7) → E (4); T (19) → Q (16); X (23) → U (20); L (11) → I (8); F (5) → C (2); N (13) → K (10); E (4) → B (1); U (20) → R (17); R (17) → O (14); Z (25) → W (22); Q (16) → N (13); I (8) → F (5); R (17) → O (14); A (0) → X (23). The full decryption recovers "THEQUICKBROWNFOX". If the key kk is unknown, decryption can be attempted via brute force by trying all possible shifts from 1 to 25 (shift 0 yields the ciphertext unchanged, and shifts beyond 25 repeat periodically). For each trial shift, the recipient applies the backward shift to the entire and checks for readable English text, such as common words or patterns resembling natural language. This exhaustive method succeeds because the cipher has only 25 nontrivial keys, making it feasible to identify the correct one quickly. Successful decryption fundamentally requires either prior knowledge of kk (shared securely between sender and recipient) or an effective means to guess or deduce it, as the cipher's simplicity relies on key secrecy.

Cryptanalysis

Breaking Methods

The Caesar cipher can be broken using a , which exploits its small key space of 25 non-trivial shifts (excluding the identity shift of 0). An attacker systematically tries each possible shift value on the , decrypting the message and checking for readability or meaningful English text, with a of O(26 * n), where n is the length of the text. Frequency analysis provides a more efficient breaking method by leveraging the preserved statistical distribution of letters in , such as English where 'E' appears approximately 12.7% of the time. The attacker identifies the most frequent letter in the and tests shifts that map it to common letters like 'E' or 'T', refining the key by examining frequencies or overall coherence. For instance, if 'X' is the most frequent letter, a shift that aligns it with 'E' is likely correct. However, frequency analysis is less effective for short ciphertexts because the limited number of letter occurrences does not provide reliable statistical distributions, making brute-force trial of the 26 possible shifts more practical and efficient. A allows direct key recovery if the attacker obtains even a single corresponding plaintext-ciphertext pair, computing the shift as the modular difference between them. This method completely compromises the cipher, as the fixed shift applies uniformly across the message. Automated tools implement these techniques, breaking Caesar ciphers in seconds on modern computers due to the exhaustive search over a mere 26 possibilities or rapid matching. Historically, cryptanalysts performed these attacks manually by inspecting short ciphertexts for patterns.

Security Limitations

The Caesar cipher's primary cryptographic weakness stems from its monoalphabetic substitution design, which applies a fixed shift to every letter, thereby preserving the distribution of letters in the that mirrors patterns. This invariance allows attackers to exploit statistical analysis, such as counts of common letters like 'E' in English, to infer the shift value without the key. Compounding this vulnerability is the cipher's extremely limited key space, consisting of only 25 non-trivial shifts (excluding the identity shift of 0), which renders it susceptible to exhaustive brute-force attacks that can test all possibilities in seconds on modern hardware. In contrast, polyalphabetic ciphers like the expand the key space exponentially through keyword lengths, significantly increasing resistance to such searches. The cipher also fails to incorporate essential principles of , as articulated by , where each letter encrypts independently without spreading the influence of a single change across multiple positions or obscuring the key's role through nonlinear transformations. Without these properties, small modifications in the produce proportionally limited changes in the , facilitating and partial recoveries. In contemporary cryptography, the Caesar cipher holds no practical value for secure communications, having been superseded by robust standards like the (AES) for symmetric encryption and RSA for asymmetric key exchange, which offer vast key spaces and proven resistance to known attacks. It persists primarily in educational contexts to illustrate basic concepts or for lightweight in non-sensitive applications, such as puzzles. Variants like the represent a modest improvement by generalizing the shift to a linear transformation c=(ap+b)mod  26c = (a p + b) \mod 26, where aa is coprime to 26, yielding up to 312 possible keys and slightly complicating . However, it remains fundamentally monoalphabetic and vulnerable to the same statistical and exhaustive methods, providing negligible security gains over the original.

References

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