Historical Background
Although named after Blaise de Vigenère, the cipher was actually first described by Giovan Battista Bellaso in 1553. Vigenère published his own version in 1586, which was a more complex cipher incorporating autokeys. However, the simpler version (which is what we now call the Vigenère cipher) became associated with his name through historical confusion.
For nearly 300 years, the cipher remained unbroken, and was widely considered unbreakable. It wasn't until the 19th century that Friedrich Kasiski published a general method for deciphering Vigenère ciphers in 1863.
Mathematical Representation
Mathematically, the Vigenère cipher can be expressed using modular arithmetic:
Encryption
Ci = (Pi + Ki mod m) mod n
Where:
- Ci is the i-th character of the ciphertext
- Pi is the i-th character of the plaintext
- K is the key
- m is the length of the key
- n is the size of the alphabet (26 for standard Latin alphabet)
Decryption
Pi = (Ci - Ki mod m + n) mod n
Where:
- Pi is the i-th character of the plaintext
- Ci is the i-th character of the ciphertext
- K is the key
- m is the length of the key
- n is the size of the alphabet (26 for standard Latin alphabet)
The Tabula Recta
The Vigenère cipher is often implemented using a table called the "tabula recta" or "Vigenère square" - a 26×26 table containing the alphabet written out 26 times in different rows, each alphabet shifted cyclically to the left compared to the previous alphabet.
To encrypt, find the row with the first letter of the key, and the column with the first letter of the plaintext. The intersection is the encrypted letter. Repeat for each letter, cycling through the key as needed.
Variants and Extensions
Several variations of the Vigenère cipher have been developed:
- Autokey Cipher: Instead of repeating the keyword, the plaintext itself is used as the key (after the initial keyword)
- Running Key Cipher: Uses a text as the key (such as a page from a book), providing an extremely long and non-repeating key
- Beaufort Cipher: A variant that uses a different formula for encryption where the key letter is subtracted from the plaintext letter
- Gronsfeld Cipher: Uses numbers instead of letters as the key
Breaking the Vigenère Cipher
Despite its historical reputation as "le chiffre indéchiffrable", the Vigenère cipher can be broken through several attacks:
1. Kasiski Examination
This method, developed by Friedrich Kasiski in 1863, looks for repeated segments in the ciphertext, which often indicate that the same plaintext was encrypted with the same part of the key. The distances between these repetitions can help determine the key length.
2. Friedman Test
Also known as the index of coincidence method, this statistical approach helps determine the key length by analyzing the frequency distribution of characters in the ciphertext.
3. Frequency Analysis
Once the key length is known, the ciphertext can be divided into groups where each group was encrypted with the same key letter. Each group can then be analyzed as a simple substitution cipher.
Security Limitations
While much stronger than simple substitution ciphers like the Caesar cipher, the Vigenère cipher is still not secure by modern standards. Its security relies heavily on the key remaining secret and being at least as long as the message. For secure encryption, modern algorithms like AES, RSA, or ECC should be used instead.
Implementation Examples
Python Implementation
def vigenere_encrypt(text, key):
result = ""
key_length = len(key)
key_as_int = [ord(k.upper()) - ord('A') for k in key]
for i, char in enumerate(text):
if char.isalpha():
# Determine if character is uppercase or lowercase
if char.isupper():
base = ord('A')
else:
base = ord('a')
# Convert to 0-25
char_as_int = ord(char) - base
# Apply Vigenère encryption formula
key_index = i % key_length
encrypted = (char_as_int + key_as_int[key_index]) % 26
# Convert back to ASCII and add to result
result += chr(base + encrypted)
else:
# Keep non-alphabetic characters as they are
result += char
return result
def vigenere_decrypt(text, key):
result = ""
key_length = len(key)
key_as_int = [ord(k.upper()) - ord('A') for k in key]
for i, char in enumerate(text):
if char.isalpha():
# Determine if character is uppercase or lowercase
if char.isupper():
base = ord('A')
else:
base = ord('a')
# Convert to 0-25
char_as_int = ord(char) - base
# Apply Vigenère decryption formula
key_index = i % key_length
decrypted = (char_as_int - key_as_int[key_index] + 26) % 26
# Convert back to ASCII and add to result
result += chr(base + decrypted)
else:
# Keep non-alphabetic characters as they are
result += char
return resultJavaScript Implementation
function vigenereEncrypt(text, key) {
let result = '';
const keyLength = key.length;
// Convert key to array of shift values (0-25)
const keyShifts = key.toUpperCase().split('').map(k =>
k.charCodeAt(0) - 65
);
// Process each character
for (let i = 0; i < text.length; i++) {
const char = text[i];
// Check if character is a letter
if (/[A-Za-z]/.test(char)) {
// Determine ASCII offset based on case
const isUpper = char === char.toUpperCase();
const base = isUpper ? 65 : 97;
// Convert to 0-25 range
const charIndex = char.charCodeAt(0) - base;
// Get key shift for this position
const keyIndex = i % keyLength;
const shift = keyShifts[keyIndex];
// Apply Vigenère encryption
const encryptedIndex = (charIndex + shift) % 26;
// Convert back to character and append
result += String.fromCharCode(base + encryptedIndex);
} else {
// Keep non-alphabetic characters as they are
result += char;
}
}
return result;
}
function vigenereDecrypt(text, key) {
let result = '';
const keyLength = key.length;
// Convert key to array of shift values (0-25)
const keyShifts = key.toUpperCase().split('').map(k =>
k.charCodeAt(0) - 65
);
// Process each character
for (let i = 0; i < text.length; i++) {
const char = text[i];
// Check if character is a letter
if (/[A-Za-z]/.test(char)) {
// Determine ASCII offset based on case
const isUpper = char === char.toUpperCase();
const base = isUpper ? 65 : 97;
// Convert to 0-25 range
const charIndex = char.charCodeAt(0) - base;
// Get key shift for this position
const keyIndex = i % keyLength;
const shift = keyShifts[keyIndex];
// Apply Vigenère decryption (add 26 to ensure positive result)
const decryptedIndex = (charIndex - shift + 26) % 26;
// Convert back to character and append
result += String.fromCharCode(base + decryptedIndex);
} else {
// Keep non-alphabetic characters as they are
result += char;
}
}
return result;
}Conclusion
The Vigenère cipher represents an important advancement in cryptographic history, moving beyond simple monoalphabetic substitution to polyalphabetic substitution. Its resistance to frequency analysis made it considerably more secure than previous ciphers, and its conceptual foundation influenced the development of many subsequent encryption methods. While no longer considered secure for modern applications, understanding the Vigenère cipher provides valuable insights into the evolution of cryptographic techniques and the fundamental principles of encryption.