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At a minimum $P(x)$ must be primitive and $f:\{0,1\}^n
\rightarrow \{0,1\}$ must be highly nonlinear and resilient of high order (correlation immune of high order plus balanced) are necessary conditions.
Nonlinearity (minimal Hamming distance of the truth table of the boolean function from affine functions), must be high for resisting linear/affine ...
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