Caesar & Vigenère Ciphers: From Frequency Analysis to Kasiski's Attack

Shifting the alphabet

The Caesar cipher, reportedly used by Julius Caesar for military correspondence, encrypts by shifting every letter of the alphabet forward by a fixed amount k: A becomes D under a shift of 3, B becomes E, and so on, wrapping from Z back to A. Decryption just shifts backward by the same k. It has exactly 25 possible keys (26 minus the useless shift of 0), which means it offers essentially no real security — a computer, or a patient human, can simply try all 25 shifts and read off which one produces English.

Vigenère: a different shift for every letter

The Vigenère cipher (16th century, long misattributed and once called "le chiffre indéchiffrable") fixes the single-shift weakness by repeating a keyword under the plaintext and using each keyword letter's own alphabet position as that position's shift.

plaintext:   A T T A C K A T D A W N
key:         L E M O N L E M O N L E
ciphertext:  L X F O P V E F R N H R    (each letter shifted by its aligned key letter)

C_i = (P_i + K_{i mod m}) mod 26,   m = key length

Because the shift changes letter by letter following the keyword, the same plaintext letter usually encrypts to different ciphertext letters at different positions, which flattens the single-alphabet letter-frequency pattern that makes Caesar so breakable — for nearly three centuries this genuinely resisted casual cryptanalysis.

Breaking Caesar: frequency analysis

English letter frequencies are famously uneven: E appears about 12.7% of the time, T about 9.1%, while J, Q, X and Z barely register. Under a Caesar shift, this entire frequency profile just slides by k positions, so an attacker measures the ciphertext's letter frequencies, finds the shift that best aligns the peak (usually the ciphertext's most common letter mapping back to E) with known English statistics, and recovers k directly — exactly the automated highlight this simulation performs on the live frequency chart.

Breaking Vigenère: finding the key length first

Vigenère resists simple frequency analysis on the whole ciphertext because 26 different shift alphabets are interleaved. The classical attack, Kasiski examination, instead looks for repeated substrings in the ciphertext; if the same plaintext fragment happens to align with the same part of the repeating key twice, it produces an identical ciphertext substring, and the distance between repeats is very likely a multiple of the key length m. Once m is known, the ciphertext splits into m independent Caesar-shifted streams (every m-th letter shares one shift), and ordinary frequency analysis breaks each one separately.

The index of coincidence: measuring 'how English' a stream looks

The index of coincidence (IC) gives a more robust, statistical way to find m without hunting for repeats. It measures the probability that two randomly chosen letters from a text are identical.

IC = Σ n_i(n_i − 1) / (N(N − 1))     n_i = count of letter i, N = total letters

random text (26 equiprobable letters):  IC ≈ 0.038
normal English text:                    IC ≈ 0.067
Vigenère split at the correct key length m: each of the m streams is single-shifted
English, so its IC lands near 0.067; split at the wrong length, the streams behave
closer to random and IC drops toward 0.038.

Trying candidate key lengths and picking the one whose split streams show the highest average IC pins down m reliably, after which the same frequency-analysis attack that breaks Caesar breaks each of the m streams in turn — the two attacks in this simulation are, in the end, the same idea applied once and then m times in parallel.

Frequently asked questions

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