> ## Documentation Index
> Fetch the complete documentation index at: https://docs.cipherinspector.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Chi-squared Test

> The Chi-squared test measures how closely the character or n-gram frequency distribution of a ciphertext matches an expected distribution.

## How it works

<Steps>
  <Step title="Decoding">
    The Chi-squared test uses decoded ciphertext content. For data encodings (Base64, Hexadecimal, Decimal, etc.), the raw data is first decoded to bytes and then converted to characters.
  </Step>

  <Step title="Ciphertext Settings">
    Chi-squared takes into consideration the settings for each ciphertext, including:

    * Ignore whitespace
    * Ignore punctuation
    * Ignore casing
    * Genericize text

    These settings are applied before frequency analysis.
  </Step>

  <Step title="N-gram Generation">
    The text is divided into n-grams (character groupings) based on the configured n-gram size:

    * **Size 1**: Single characters (default)
    * **Size 2**: Bigrams (pairs of characters)
    * **Size 3**: Trigrams (triplets of characters)
    * And so on...

    N-grams can be generated using either:

    * **Sliding window**: Overlapping n-grams (ABCD → AB, BC, CD)
    * **Block mode**: Non-overlapping n-grams (ABCD → AB, CD)
  </Step>

  <Step title="Frequency Calculation">
    The frequency of each n-gram is counted. The n-grams analyzed depend on the comparison mode:

    * **English mode** (n-gram size 1 only): Only alphabetic characters (A-Z) are counted; all other characters are filtered out.
    * **Ciphertext mode**: All n-grams are counted, including those with letters, numbers, symbols, and whitespace.
  </Step>

  <Step title="Chi-squared Calculation">
    The Chi-squared statistic is calculated using the formula:

    **X² = Σ(Oᵢ - Eᵢ)² / Eᵢ**

    Where:

    * Oᵢ = Observed count of n-gram i
    * Eᵢ = Expected count of n-gram i

    The expected count for each n-gram is calculated as: (expected percentage / 100) × total n-gram count
  </Step>
</Steps>

## Chi-squared Settings

### Comparison Mode

#### English Frequencies (Default)

Compares the ciphertext against standard English letter frequencies. **Only available for n-gram size 1** (single characters), since we only have reference frequencies for individual letters.

Only alphabetic characters (A-Z) are analyzed; all other characters are filtered out before comparison.

#### Another Ciphertext

Compares the ciphertext against the n-gram frequency distribution of a selected "base" ciphertext. Works with any n-gram size.

After applying the ciphertext settings (ignore whitespace, ignore punctuation, etc.), all remaining n-grams are analyzed.

When this mode is selected, you choose which ciphertext serves as the baseline for comparison.

### N-gram Settings

#### N-gram Size

Controls how many characters are grouped together for frequency analysis:

* **1** (default): Analyze single character frequencies
* **2**: Analyze bigram (two-character) frequencies
* **3**: Analyze trigram (three-character) frequencies
* **4+**: Analyze larger n-gram frequencies

Larger n-gram sizes can reveal patterns in polyalphabetic ciphers or detect repeated sequences.

**Note**: English frequency comparison is only available for n-gram size 1.

#### N-gram Mode

* **Sliding Window**: Creates overlapping n-grams. For "ABCD" with size 2: AB, BC, CD
* **Block**: Creates non-overlapping n-grams. For "ABCD" with size 2: AB, CD

Sliding window produces more n-grams and may reveal more patterns, while block mode treats the text as discrete chunks.

### Display Type

#### Table

Shows a summary for each ciphertext including:

* Chi-squared score
* Score interpretation
* Number of n-grams analyzed

#### Graph

Displays a bar chart showing each n-gram's contribution to the overall Chi-squared score:

* **English mode**: Letters arranged by English frequency (most common to least common)
* **Ciphertext mode / N-gram mode**: N-grams arranged by observed frequency

## Score Interpretation

| Score Range | Interpretation        |
| ----------- | --------------------- |
| 0 - 30      | Excellent match       |
| 30 - 50     | Very good match       |
| 50 - 100    | Good match            |
| 100 - 150   | Moderate deviation    |
| 150 - 300   | Significant deviation |
| 300+        | Very different        |

Lower scores indicate that the n-gram frequency distribution more closely matches the expected distribution.

## Practical Application

Chi-squared analysis can be leveraged to:

* **Identify plaintext**: A Chi-squared score near 0-50 against English frequencies suggests the text may be plaintext or a simple transposition cipher.
* **Detect substitution ciphers**: Higher scores (100-300) often indicate substitution ciphers.
* **Analyze polyalphabetic ciphers**: Using bigram or trigram analysis can reveal patterns not visible in single-letter analysis.
* **Compare ciphertexts**: Use ciphertext comparison mode to determine if two encrypted messages share similar n-gram distributions.
* **Detect repeated patterns**: Larger n-gram sizes can identify repeated sequences in ciphertexts.

## Caveats

* **English mode** only supports n-gram size 1 (single letters) because we don't have reference frequencies for English bigrams, trigrams, etc.
* Short ciphertexts may produce unreliable results, especially with larger n-gram sizes.
* Larger n-gram sizes produce fewer total n-grams, which may reduce statistical significance.
* When comparing against another ciphertext, ensure the base ciphertext is long enough to provide representative n-gram frequencies.