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How it works

The Kolmogorov-Smirnov (K-S) test is a statistical method that compares two probability distributions by measuring the maximum vertical distance between their cumulative distribution functions (CDFs).
1

Character Frequency Counting

The widget first counts the frequency of each character (or n-gram) in your ciphertext. This creates an observed distribution of how often each character appears.
2

Reference Distribution

A reference distribution is selected based on your comparison mode:
  • Uniform: Equal probability for all observed characters
  • English: Standard English letter frequencies (e.g., E at 12.7%, T at 9.06%)
  • Ciphertext: The character distribution from another ciphertext you select
3

CDF Construction

Both the observed and reference distributions are converted into cumulative distribution functions. A CDF shows, for each character, the total probability of that character and all characters before it.
4

D-Statistic Calculation

The D-statistic is calculated as the maximum absolute difference between the two CDFs at any point. A smaller D-statistic indicates the distributions are more similar.
5

P-Value Calculation

The p-value represents the probability that the observed difference could occur by chance. Higher p-values suggest the distributions match well; lower p-values indicate significant differences.

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Comparison Modes

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Uniform Distribution

Compares your ciphertext against a uniform distribution where every character has equal probability. This is useful for detecting whether encryption has produced evenly distributed output.

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English Frequencies

Compares your ciphertext against standard English letter frequencies. Only alphabetic characters (A-Z) are analyzed. This helps identify if text resembles natural English.

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Another Ciphertext

Compares your ciphertext against the character distribution of a selected base ciphertext. Useful for determining if two ciphertexts were encrypted using similar methods or share statistical properties.

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Display Modes

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Score View

Displays a table with:
  • D-statistic: The maximum difference between CDFs (0 to 1 scale)
  • P-value: Statistical significance (color-coded for quick interpretation)
  • Interpretation: Human-readable assessment of the result
  • Sample size: Number of characters or n-grams analyzed

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CDF Graph View

Displays an interactive chart showing:
  • Solid lines: Observed CDF from your ciphertext
  • Dashed lines: Expected CDF from the reference distribution
  • Dotted vertical line: Location of maximum difference (D-statistic)

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N-gram Settings

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N-gram Size

Instead of analyzing single characters, you can group characters into n-grams:
  • 1: Single characters (default)
  • 2: Bigrams (pairs like “TH”, “HE”)
  • 3: Trigrams (triplets like “THE”, “AND”)
Larger n-grams capture patterns in character sequences but require longer texts for meaningful analysis.

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N-gram Mode

  • Sliding Window: Overlapping n-grams (ABCD → AB, BC, CD)
  • Block: Non-overlapping n-grams (ABCD → AB, CD)

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P-Value Interpretation

P-value RangeInterpretation
> 0.10Distributions match well
0.05 - 0.10Slight deviation
0.01 - 0.05Significant deviation
< 0.01Very different distributions

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Practical Applications

The Kolmogorov-Smirnov test can be used to:
  • Determine if ciphertext has uniform byte distribution (suggesting strong encryption)
  • Identify if plaintext resembles natural English
  • Compare multiple ciphertexts to detect similar encryption methods
  • Analyze whether a substitution cipher preserves frequency patterns

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Caveats

  • English frequency comparison only analyzes alphabetic characters; non-alphabetic characters are filtered out
  • For n-gram sizes greater than 1, English frequency comparison falls back to uniform distribution (no reference English n-gram frequencies available)
  • Very short texts may produce unreliable p-values due to small sample sizes
  • The test measures overall distribution similarity, not specific character mappings