  We study the problem of privacy amplification with an active adversary in the
information theoretic setting. In this setting, two parties Alice and Bob start
out with a shared $n$-bit weak random string $W$, and try to agree on a secret
random key $R$ over a public channel fully controlled by an active and
unbounded adversary. Typical assumptions are that these two parties have access
to local private uniform random bits. In this paper we seek to minimize the
requirements on the local randomness used by the two parties.
  We make two improvements over previous results. First, we reduce the number
of random bits needed for each party to $\Theta(\ell+\log n)$, where $\ell$ is
the 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 the
problem to the case where the two parties only have local weak random sources
instead of truly uniform random bits. We show that when each party has a local
weak random source with min-entropy $> n/2$, there is an efficient privacy
amplification protocol that works nearly as good as if the two parties have
access to local uniform random bits. Next, in the case where each party only
has a weak random source with arbitrarily linear min-entropy, we give an
efficient privacy amplification protocol where we can achieve security
parameter up to $\Omega(\log k)$. Our results give the first protocols that
achieve privacy amplification when each party only has access to a local weak
random source.
