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The second-order non-linearity, and the best quadratic approximations, of Boolean functions are studied in this paper. The authors prove that cubic functions within the Maiorana-McFarland class achieve very high second order non-linearity, which is close to an upper bound that was recently proved by Carlet et al., and much higher than the second order non-linearity obtained by other known constructions. The structure of the cubic Boolean functions considered allows the efficient computation of (a subset of) their best quadratic approximations.
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