A hidden channel also changes a distribution.

If ordinary encoders choose among representations unevenly, mapping secret bits directly to choices can flatten that pattern. A better selection process tries to preserve the frequencies expected from innocent data.

SYNTHETIC CATEGORICAL MODEL / NO CLAIM OF PERFECT SECURITY

STATISTICAL LAB

Watch evidence accumulate

Eight abstract choices stand in for masks, segment plans, or another legal encoder decision. The model does not produce operational steganography.

Predict first

Naive hiding maps three secret bits straight onto the eight choices, making every choice equally likely. Will that look like ordinary encoder behaviour?

Yes — every choice is legal No — it stands out

NORMAL COVER

Expected encoder behavior

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NAIVE SECRET ENCODING

Three bits map directly to eight choices

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DISTRIBUTION-SHAPED SELECTION

Secret-derived draws follow the cover CDF

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WHAT THE OBSERVER EXPECTS

Choices 0-7 occur with fixed, unequal probabilities in the synthetic cover model.

WHAT NAIVE HIDING CHANGES

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WHAT SHAPING PRESERVES

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WHAT AN ANALYST CAN TEST

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THE FOUR QUESTIONS

Ask four questions of the channel.

WHAT THE SCANNER SEES

Nothing unusual. Every sample is a valid symbol that decodes normally — the channel lives in which legal choice was made, not in what was encoded.

WHAT CHANGED UNDERNEATH

The frequency with which each of the eight choices is selected, measured across many symbols rather than inside any one of them.

WHAT A SECOND READER CAN RECOVER

The secret bits, by inverting the same selection model. The recipient has to share the cover model, which is the real cost of this approach.

HOW AN ANALYST CAN NOTICE

Test observed frequencies against expected ones. Naive mapping flattens the distribution and fails a goodness-of-fit test; shaped selection is built to pass this particular test, which is not the same as being undetectable.

MODEL LIMIT

Matching one visible distribution does not prove undetectability.

The shaped samples are drawn from the same eight-choice probability model, so this particular chi-square test often cannot separate them from normal samples. A real system has dependencies, side channels, imperfect models, and more capable analysts.

Conceptual basis: model-based and distribution-preserving ideas associated in this repository with Sallee, grey-box steganography, Meteor, Discop, minimum-entropy coupling, and set shaping. These references remain [2ND] unless upgraded in the research record. No named algorithm is reproduced here.

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