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We study the problem of privacy amplification with an active adversary in theinformation theoretic setting. In this setting, two parties Alice and Bob startout with a shared $n$-bit weak random string $W$, and try to agree on a secretrandom key $R$ over a public channel fully controlled by an active andunbounded adversary. Typical assumptions are that these two parties have accessto local private uniform random bits. In this paper we seek to minimize therequirements on the local randomness used by the two parties.  We make two improvements over previous results. First, we reduce the numberof random bits needed for each party to $\Theta(\ell+\log n)$, where $\ell$ isthe security parameter, as long as $W$ has min-entropy $n^{\Omega(1)}$.Previously, the best known result needs to use $\Theta((\ell+\log n)\log n)$bits. Our result is also asymptotically optimal. Second, we generalize theproblem to the case where the two parties only have local weak random sourcesinstead of truly uniform random bits. We show that when each party has a localweak random source with min-entropy $> n/2$, there is an efficient privacyamplification protocol that works nearly as good as if the two parties haveaccess to local uniform random bits. Next, in the case where each party onlyhas a weak random source with arbitrarily linear min-entropy, we give anefficient privacy amplification protocol where we can achieve securityparameter up to $\Omega(\log k)$. Our results give the first protocols thatachieve privacy amplification when each party only has access to a local weakrandom source.

On the Problem of Local Randomness in Privacy Amplification with an Active Adversary