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I was answering this question Computing Logarithm using homomorphic encryption and I came up with a solution if you had encryptions of the bits of the number that you wanted to take the log of. But in that situation all they have is a fully homomorphic encryption of the number itself.

I was curious if there was a way to compute encryptions of the bits from an encryption of the number. i.e. is there some $f$ such that $f(E(x)) = (E(x_1),E(x_2),...,E(x_n))$ where $x_i$ is the $i$th bit of x?

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