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For a plaintext alphabet of size $m$ and entropy rate $h$ in bits per symbol, put $d={\log }_{2}m-h$. In the random-cipher approximation, the unicity distance scale iswhere $H(K)$ is cryptographic key Shannon entropy. Uniform cryptographic keys on a finite set $\mathcal{K}$ give $H(K)={\log }_2|\mathcal{K}|$. For correlated plaintext, its one-symbol Shannon entropy cannot replace the entropy rate without justification.^[ref]

For example, a 40-bit uniform cryptographic key and redundancy two bits per symbol suggest about twenty symbols under this model. This estimate does not bound computation time or guarantee unique recovery for each ciphertext. At zero redundancy it supplies no finite threshold; a probability-preserving source symmetry can also invalidate the random-cipher uniqueness intuition.

A source symmetry is a transformation of possible plaintexts that preserves their probability distribution. If different cryptographic keys induce candidate decryptions related by such a symmetry, source statistics cannot distinguish those keys, even when the ciphertext becomes arbitrarily long.

Let independent plaintext bits $X_i$ be fair, and reuse one independent fair key bit $K$ through $C_i=X_i\oplus K$. Flipping every plaintext bit preserves the source law. For either key value, $\mathbb{P}(C_1=c_1,\dots ,C_n=c_n\mid K)=2^{-n}$, so observing any finite ciphertext leaves the two key probabilities equal. This example has only two keys but no finite unicity distance; redundancy assumptions are essential to a source-based threshold estimate.^[ref]

Unicity distance is the ciphertext-length scale at which a specified cipher and plaintext model are expected to leave essentially one plausible cryptographic key. Claude Shannon obtained an approximate threshold from the sharp decrease of key equivocation in a random-cipher model; it is not a universal length guaranteeing zero residual conditional entropy for every ciphertext.^[ref]

For a realized ciphertext, uniqueness depends on which candidate decryptions the plaintext model allows. A probability distribution giving every finite plaintext positive mass need not exclude every wrong cryptographic key at any finite length. A source symmetry can even preserve ambiguity forever. The source redundancy in cryptanalysis estimate therefore needs its source and cipher assumptions.