Vigenere Cipher Decryption: How to Find the Unknown Key

The Kasiski examination finds repeated trigrams in ciphertext and measures distances between occurrences. For 'JSU SUM VWV MPH', distances of 100, 117, and 80 reveal a key length of 4. The ciphertext is divided into 4 separate monoalphabetic ciphers. Frequency analysis on each group reveals individual keys. Testing shifts on 'LWGF' with key length 4 reveals the key 'JIED' and plaintext 'JULIUS CAESAR'.

Kasiski's method is a cryptanalysis technique for breaking Vigenère cipher by identifying repeated ciphertext patterns (n-grams) and calculating the distances between them; the key length is determined by finding the greatest common divisor of these distances, as repeated plaintext segments encrypted with the same key segment will produce identical ciphertext patterns when their distance is a multiple of the key length.

The Kasiski method is a cryptanalysis technique for breaking Vigenere ciphers by identifying repeated ciphertext segments (bigrams or trigrams) and calculating the distances between them; the key length is determined by finding the greatest common divisor of these distances, after which the key can be recovered using frequency analysis on the separated plaintext segments.

The Kasiski examination exploits the repetition of key patterns in polyalphabetic ciphers. By identifying repeated patterns in the ciphertext (e.g., HNB, KGA, PNP) and measuring the distances between their occurrences, cryptanalysts can find common factors. These factors reveal the key length. Once the key length is known, the ciphertext can be divided into separate monoalphabetic ciphers, each solvable through frequency analysis.

Charles Babbage developed the Kasiski examination in 1824 to break polyalphabetic ciphers by analyzing repeated sequences in ciphertext. Babbage observed that repeated sequences likely resulted from the same plaintext sequence encrypted with the same key portion. By measuring distances between repeated sequences and finding their greatest common divisor, cryptanalysts determine the key length. Once key length is known, the ciphertext is divided into groups, each encrypted with a different alphabet. Each group is then analyzed using frequency analysis, comparing letter frequencies to known language statistics. The shifts for each key position are determined by matching frequency patterns to expected patterns in the language.

This section introduces the One-Time Pad (OTP), a cipher that achieves perfect secrecy when used correctly. The video explains that OTP uses a truly random pad of the same length as the plaintext, generated using a random number generator (RNG). The pad is XORed with the plaintext to produce ciphertext. The section emphasizes that OTP is unbreakable because the key length equals the message length and the key is truly random, providing perfect secrecy. The video concludes by noting that the more random the pad, the more secure the cipher becomes.

The one-time pad achieves perfect secrecy by using uniform random keys as long as the message. Encryption computes ciphertext as m XOR k, and decryption computes m = c XOR k. Using Bayes' theorem, the posterior probability P(m|c) equals the prior probability P(m) because P(c|m) = 1/2^n for any m and c. This mathematical property ensures the ciphertext reveals nothing about the plaintext, satisfying the perfect secrecy definition.

The One-Time Pad (OTP) achieves perfect security when: (1) The key is truly random and uniformly distributed, (2) The key is as long as the message, (3) The key is used only once (hence 'one-time'), and (4) The key is kept completely secret. Under these conditions, the ciphertext provides no information about the plaintext, regardless of the adversary's computational power. This is the only encryption scheme that achieves perfect security, but its impracticality for large-scale use makes it suitable only for very small, highly sensitive communications.

The one-time pad achieves perfect secrecy by XORing plaintext with a random keystream that is as long as the message and never reused. The key requirements are: the keystream must be truly random, longer than the message, and used only once. When these conditions are met, the ciphertext provides no information about the plaintext—even with unlimited computational power, an attacker cannot distinguish the correct plaintext from any other possible message of the same length. This mathematical proof of security, established by Claude Shannon, demonstrates that the one-time pad is unbreakable under these conditions.

The One-Time Pad (Vernam cipher) achieves perfect secrecy by XORing each plaintext bit with a random key bit of the same length. Since the key is truly random and as long as the plaintext, any ciphertext bit reveals no information about the corresponding plaintext bit—both possibilities (0 or 1) remain equally likely regardless of the ciphertext. This was used historically for secure communications between Washington and Moscow during the Cold War, though practical limitations include the need for key distribution equal to the message length.

Since Caesar cipher uses shifts 0-25, there are only 26 possible shift values. To decrypt unknown ciphertext, generate all 26 possible decryptions by trying each shift value. The correct plaintext will be among these candidates. For statistical cryptanalysis, calculate the frequency distribution of each candidate decryption and compare it to expected Spanish frequencies (e.g., 'e' ~12.53%, 'a' ~13.68%, 'w' ~0.01%). The chi-squared distance measures distribution similarity: χ² = Σ((observed - expected)² / expected). The candidate with minimum chi-squared value is the correct decryption.

Python enables automated frequency analysis by counting letter occurrences in encrypted text, sorting by frequency, and comparing against known English distributions. Initial mappings are refined iteratively by examining decrypted partial text for recognizable words and patterns. Diagraph analysis extends this to letter pairs (most common is 'th'), resolving ambiguities in shorter texts. This combination of automation and human pattern recognition demonstrates how computational tools accelerate cryptanalysis while human intuition validates results.

This lecture covers the fundamental principles of cryptanalysis for classical ciphers, including four attack models (cipher text only, known plain text, chosen plain text, and chosen cipher text) with increasing strength, and demonstrates how statistical frequency analysis of English language letter distributions (with E being the most common at ~12%) enables decryption of monoalphabetic ciphers like shift and affine ciphers, while polyalphabetic ciphers require determining key length through Kasiski examination and index of coincidence testing before applying mutual index of coincidence for key recovery; hill ciphers resist frequency analysis due to their linear transformation properties but can be broken through chosen plain text attacks using matrix inversion.

Cryptanalysis breaks ciphers through systematic attacks. Three attack types exist: chosen plaintext (easiest, attacker generates pairs), known plaintext (partial knowledge), and ciphertext-only (hardest). Brute force tests all keys but fails against large key spaces. Classical cryptanalysis relies on language statistics: English letters follow predictable frequencies ('e' most common), and bigrams like 'th' dominate. Ciphers flatten frequencies for security—transposition preserves unigrams, monoalphabetic substitution scrambles them, and polyalphabetic ciphers like Vigenère approach uniform randomness. Manual breaking involves counting unigrams, identifying frequent letters as likely 'e', analyzing word patterns, and rearranging transposed text to reveal plaintext.

Statistical cryptanalysis exploits inherent patterns in human language to break ciphers without prior knowledge of the plaintext. The index of coincidence measures the probability that two randomly selected letters from a text are identical—random text yields approximately 0.038, while natural English text reaches 0.067-0.072. This metric works because some letters appear more frequently than others in natural language. By decrypting ciphertext with various configurations and measuring the resulting index of coincidence, cryptanalysts can identify promising settings. Higher scores indicate the output is more likely to be meaningful text rather than random noise.

Classical ciphers like Caesar cipher and Enigma use single-character substitution with one-to-one mappings, making them vulnerable to frequency analysis. Modern block ciphers process larger blocks (e.g., 128 bits) simultaneously, fundamentally changing security properties. This evolution addresses the critical weakness of single-character substitution where each character maps to only one possible ciphertext value.

Transposition ciphers rearrange plaintext letters according to a permutation pattern without changing the letters themselves, but are devastatingly vulnerable to known plaintext attacks. Electromechanical devices like the Enigma machine represented a major advancement, using rotors - rotating disks with alphabets on both sides connected by random wiring. Multiple rotors created complex substitutions, with the reflector reversing the electrical path for encryption/decryption. The key consisted of rotor positions, ring settings, and plugboard connections. The breaking of Enigma during World War II had enormous strategic importance, allowing the Allies to anticipate German movements. Modern symmetric encryption algorithms are based on SP networks (Substitution-Permutation networks) that alternate between substitution and permutation layers.

Cryptography evolved from simple substitution ciphers like Caesar's letter-shifting technique to complex mechanical systems. Substitution ciphers preserve letter frequencies, making them vulnerable to frequency analysis. Permutation ciphers rearrange letter positions instead. The Enigma machine represented a major advancement using multiple rotors that changed substitution mappings dynamically with each keystroke. Despite its sophistication, Enigma had a critical weakness: letters could never encrypt to themselves. Breaking Enigma required significant computational effort, achieved by Alan Turing and colleagues at Bletchley Park through automated methods during WWII.

This extensive section traces the complete historical and technical evolution of cryptography from ancient times to modern symmetric algorithms. It begins with the Caesar shift cipher, which shifts letters by a fixed amount but is vulnerable due to only 25 possible keys. The substitution cipher improves security with approximately 4×10^56 keys but remains insecure because it fails to hide statistical patterns in English text, allowing frequency analysis attacks. The Vigenère cipher introduces polyalphabetic substitution using a repeating keyword, making frequency analysis ineffective. However, it can still be broken by identifying repeated ciphertext sequences that reveal keyword length, then applying frequency analysis to individual groups. The section then examines the mechanization of cryptography during World War II, focusing on the German Enigma machine, which used rotors, a reflector, and a plugboard to create complex transformations where identical plaintext letters encrypted differently each time. Despite its sophistication, it was broken at Bletchley Park primarily through exploiting operational errors rather than mathematical weaknesses. The modern section covers symmetric cryptography: stream ciphers generate pseudo-random byte streams using a key seed and XOR with plaintext; block ciphers process fixed-size blocks (AES uses 128-bit blocks). Critical security requirements include never reusing keys, using unique initialization vectors, implementing message authentication, and avoiding arbitrary seeking. Detailed technical coverage of AES includes Substitute bytes, Shift rows, Mix columns, and Add round key operations across multiple rounds. Block cipher modes explored include ECB (revealing patterns), CTR (enabling random access), and GCM (providing authentication).

This section traces the evolution from mechanical ciphers to modern computer-based encryption. It explains why mechanical ciphers like Enigma became inadequate: they require physical hardware, are difficult to implement in computer systems, and their security strength is limited by mechanical complexity. DES (Data Encryption Standard), developed by IBM in 1977, was the first computer-based encryption algorithm using a 64-bit key and 64-bit blocks with 16 rounds of encryption. The lecture details DES's internal structure including initial permutation, F-function processing with S-boxes, and the 16-round structure. DES was designed to resist known cryptanalytic attacks but was later broken by differential cryptanalysis in 1998. The section concludes with modern standards: SEED (Korean National Standard) and AES (Advanced Encryption Standard), both using 128-bit keys and 128-bit blocks with different internal structures (Feistel vs. SPN), representing the evolution from DES's 64-bit key to modern 128-bit keys for enhanced security.