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How it works

1

Encoding

Index of coincidence (abbreviated as “IC”) will be different based on the character encoding of your input.For all data encodings, the data is first decoded to a Uint8Array and then converted to a string, thus producing a Latin-1 (ISO/IEC 8859-1) string (8-bit “extended” ASCII table). This step takes precedence over any other ciphertext settings.Display encodings are not decoded before analysis and are analyzed as-is.
2

Ciphertext Settings

IC takes into consideration the settings for each ciphertext. These settings currently include:
  • Ignore whitespace
  • Ignore punctuation
  • Ignore casing
  • Genericize text
For data encodings (base64, hex, octal, decimal, etc.) the ciphertext is first decoded to Latin-1 before the toggles are applied.
3

IC Calculations

After all of the above steps are performed, the IC calculations are executed and displayed.

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IC formula

This widget uses the following formula for calculating index of coincidence:
IC = (Σ(n_i × (n_i - 1))) / (N × (N - 1))
Where:
  • n_i = frequency of n-gram i (or character i when n-gram size = 1)
  • N = total number of n-grams (or characters)
  • Σ = sum over all unique n-grams/characters

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Periodic IC formula

Periodic IC is different based on the mode.

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N-gram mode: Block

For this mode, the process is:
  1. Generate n-grams from the entire text
  2. Group n-grams by their index modulo the period:
Group_k = {ngram_i where i % p = k} for k = 0, 1, ..., p-1
  1. Calculate IC for each group using the basic IC formula
  2. Average the IC values across all groups:
IC(p) = (1/p) × Σ(IC(Group_k)) for k = 0 to p-1

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N-gram mode: Sliding window

For this mode, the process is:
  1. Slide a window of size ngramSize across the text
  2. For each window:
    • Generate n-grams within the window
    • Group by index modulo period (same as block mode)
    • Calculate and average group ICs
  3. Average the window ICs:
IC(p) = (1/num_windows) × Σ(IC_window) for all windows

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Index of Coincidence Settings

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N-grams and sliding window vs. block analysis

IC can be performed on n-grams where n >= 1. For n-grams > 1, it is important to understand the difference between sliding window and block analysis.

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N-gram mode: Sliding window

Sliding window analysis “slides” across the ciphertext to create n-grams. For the text Hello: 
He: 1
el: 1
ll: 1
lo: 1
Notice that characters at a given index will appear at least twice for n > 1.

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N-gram mode: Block analysis

Block analysis evaluates your n-grams as non-overlapping chunks. For Hello: 
He: 1
ll: 1
Notice that the final character, o, is not present. When using block analysis, beware of missing data. The ciphertext length (after all toggles are applied) must be divisible by your n-gram size for all characters to be represented in the IC!

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Graph vs. Table

There are two display options for IC.

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Graph (Periodic analysis)

Shows a line chart, where each line is the periodic IC of a ciphertext. The height represents the IC, and the horizontal axis represents how far into the ciphertext the measurement is.

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Table

Shows a table of values, with a single IC value for each ciphertext. This is the IC for the entire ciphertext.

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Max Period

The max period is used in the graph to determine how far the x-axis will go. This helps with scaling the data in case characters are of varying lengths.

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Show Average Lines

In the periodic analysis graph, shows a dotted line for each ciphertext indicating the average line.

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Practical Application

IC can be leveraged to:
  • Determine if a cipher is periodic or aperiodic.
  • Discover key lengths of periodic ciphers, such as Vigenere with a repeat key.
  • Compare the periods of two or more ciphers.

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Caveats

  • Noisy results may mislead you.
  • Autokey is a sufficient way to get rid of periodic spikes in IC.
  • Coincidences in sufficiently short text may also mislead you.
  • Some languages have similar IC patterns. If you don’t know the language of the plaintext, you may be misled.