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Which is the smallest, cyclic in 3 directions, consistent structure of random values which can be hidden at the adversaries machine? (some comparison)

Something half baked. Let $s$ be some integer, multiple of $3$, in the order of $2^{50}$. Let the set be triplets $(x,y,z)\in[0,s)^3$, with $N=s^3\approx2^{150}$ elements. Define $$f(x,y,z)=\begin{...
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