Abstract
It has recently been shown that when m > 1/2n - 1, the nonlinearity N f of an mth-order correlation immune function f with n variables satisfies the condition of N f ≤ 2n-1 - 2m, and that when m > 1/2n - 2 and f is a balanced function, the nonlinearity satisfies N f ≤2n-1 - 2m+1. In this work we prove that the general inequality, namely N f ≤ 2n-1 - 2m, can be improved to N f ≤ 2n-1 - 2m+1 for m ≥ 0.6n - 0.4, regardless of the balance of the function. We also show that correlation immune functions achieving the maximum nonlinearity for these functions have close relationships with plateaued functions. The latter have a number of cryptographically desirable properties.
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Zheng, Y., Zhang, XM. (2001). Improved Upper Bound on the Nonlinearity of High Order Correlation Immune Functions. In: Stinson, D.R., Tavares, S. (eds) Selected Areas in Cryptography. SAC 2000. Lecture Notes in Computer Science, vol 2012. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44983-3_19
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DOI: https://doi.org/10.1007/3-540-44983-3_19
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