Is there a better than linear lower bound for factoring and discrete log? Are there any references that provide details about circuit lower bounds for specific hard problems arising in cryptography such as integer factoring, prime/composite discrete logarithm problem and its variant over group of points of elliptic curves (and their higher dimensional abelian varieties) and the general hidden subgroup problem?

Specifically do any of these problems have more than a linear complexity lower bound?
@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.