Is there a better than linear lower bound for factoring and discrete log?
@Suresh: following your advice, here is my "answer". The status of circuit lower bounds is quite depressing. Here are the "current records":


  $4n-4$ for circuits over $\{\land,\lor,\neg\}$, and $7n-7$
for circuits over $\{\land,\neg\}$ and $\{\lor,\neg\}$
  computing $\oplus_n(x)=x_1\oplus x_2\oplus\cdots\oplus x_n$;
Redkin (1973). These bounds are tight.

 $5n-o(n)$ for circuits over the basis with all
  fanin-2 gates, except the parity and its negation; Iwama and
  Morizumi (2002).

  $3n-o(n)$ for general circuits over the basis
  with all fanin-2 gates; Blum (1984). Arist Kojevnikov and Sasha Kulikov from Petersburg have found a simpler proof of a
$(7/3)n-o(1)$ lower bound. The advantage of their proof is its simplicity, not numerical. Later they gave a simple proof of $3n-o(1)$ lower bound for general circuits (all fanin-2 gates are allowed). Albeit for very complicated functions - affine dispersers. Papers are online  here.

 $n^{3-o(1)}$ for formulas over
  $\{\land,\lor,\neg\}$; Hastad (1998).

  $\Omega(n^2/\log n)$ for general fanin-$2$ formulas,
$\Omega(n^2/\log^2 n)$ for deterministic branching programs, and
$\Omega(n^{3/2}/\log n)$ for nondeterministic branching programs;
Nechiporuk~(1966).


So, your question "Specifically do any of these problems have more than a linear complexity lower bound?" remains widely open (in the case of circuits). My appeal to all young researchers: go forward, these "barriers" are not unbreakable! But try to think in a "non-natural way", in the sense of Razborov and Rudich.