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We present novel and sharp lower bounds for higher load moments in theclassical problem of mapping $M$ balls into $N$ bins by $q$-universal hashing,specialized to the case when $M=N$. As a corollary we prove a tight counterpartfor the result about min-entropy condensers due to Dodis, Pietrzak and Wichs(CRYPTO'14), which has found important applications in key derivation. Itstates that condensing $k$ bits of min-entropy into a $k$-bit string$\epsilon$-close to almost full min-entropy (precisely $k-\log\log(1/\epsilon)$ bits of entropy) can be achieved by the use of$q$-independent hashing with $q= \log(1/\epsilon)$. We prove that when given asource of min-entropy $k$ and aiming at entropy loss $\ell = \log\log(1/\epsilon) - 3$, the independence level $q=(1-o(1))\log(1/\epsilon)$ isnecessary (for small values of $\epsilon$), which almost matches the positiveresult. Besides these asymptotic bounds, we provide clear hard bounds in termsof Bell numbers and some numerical examples. Our technique is based on anexplicit representation of the load moments in terms of Stirling numbers, someasymptotic estimates on Stirling numbers and a tricky application of thePaley-Zygmund inequality. \keywords{ min-entropy condensers, key derivation,balls and bins hashing, anti-concentration inequalities }

Lower bounds on $q$-wise independence tails and applications to min-entropy condensers