  Traditional minwise hashing (MinHash) requires applying $K$ independent
permutations to estimate the Jaccard similarity in massive binary (0/1) data,
where $K$ can be (e.g.,) 1024 or even larger, depending on applications. The
recent work on C-MinHash (Li and Li, 2021) has shown, with rigorous proofs,
that only two permutations are needed. An initial permutation is applied to
break whatever structures which might exist in the data, and a second
permutation is re-used $K$ times to produce $K$ hashes, via a circulant
shifting fashion. (Li and Li, 2021) has proved that, perhaps surprisingly, even
though the $K$ hashes are correlated, the estimation variance is strictly
smaller than the variance of the traditional MinHash.
  It has been demonstrated in (Li and Li, 2021) that the initial permutation in
C-MinHash is indeed necessary. For the ease of theoretical analysis, they have
used two independent permutations. In this paper, we show that one can actually
simply use one permutation. That is, one single permutation is used for both
the initial pre-processing step to break the structures in the data and the
circulant hashing step to generate $K$ hashes. Although the theoretical
analysis becomes very complicated, we are able to explicitly write down the
expression for the expectation of the estimator. The new estimator is no longer
unbiased but the bias is extremely small and has essentially no impact on the
estimation accuracy (mean square errors). An extensive set of experiments are
provided to verify our claim for using just one permutation.
