An NP-complete variant of factoring. Arora and Barak's book presents factoring as the following problem:

$\text{FACTORING} = \{\langle L, U, N \rangle \;|\; (\exists \text{ a prime } p \in \{L, \ldots, U\})[p | N]\}$

They add, further in Chapter 2, that removing the fact that $p$ is prime makes this problem NP-complete, although this is not linked to the difficulty of factoring numbers.  It looks there can be a reduction from SUBSETSUM, but I got stuck finding it.  Any better luck around here?

EDIT March 1st: The bounty is for $NP$-completeness proof using deterministic Karp (or Cook) reduction.
This is not quite an answer, but it's close. The following is a proof that the problem is NP-hard under randomized reductions.

There's an obvious relation to subset sum which is: suppose you know the factors of $N$: $p_1$, $p_2$, $\ldots$, $p_k$. Now, you want to find a subset $S$ of $p_1$ $\ldots$ $p_k$ such that 

$$\displaystyle \log L \leq \sum_{p_i \in S} \log p_i \leq \log U.$$

The problem with trying to use this idea to show the problem is NP-hard is that if you have a subset-sum problem with numbers $t_1$, $t_2$, $\ldots$, $t_k$,
you can't necessarily find primes in polynomial time such that $\log p_i \propto t_i$ (where by $\propto$, I mean approximately proportional to). This is a real problem because, since subset-sum is not strongly NP-complete, you need to find these $\log p_i$ for large integers $t_i$. 

Now, suppose we require that all the integers $t_1$ $\ldots$ $t_k$ in a subset sum problem are between $x$ and $x(1+1/k)$, and that the sum is approximately $\frac{1}{2}\sum_i t_i$. The subset sum problem will still be NP-complete, and any solution will be the sum of $k/2$ integers. We can change the problem from integers to reals if we let $t&#39;_i$ be between $t_i$ and $t_i+\frac{1}{10k}$, and instead of requiring the sum to be exactly $s$, we require it to be between $s$ and $s + \frac{1}{10}$. We only need to specify our numbers to around $4 \log k$ more bits of precision to do this. Thus, if we start with numbers with $B$ bits, and we can specify real numbers $\log p_i$ to approximately $B + 4 \log k$ bits of precision, we can carry out our reduction.

Now, from wikipedia (via Hsien-Chih's comment below), the number of primes between $T$ and $T+ T^{5/8}$ is $\theta(T^{5/8}/\log T)$, so if you just choose numbers randomly in that range, and test them for primality, with high probability get a prime in polynomial time. 

Now, let's try the reduction. Let's say our $t_i$ are all $B$ bits long. If we take $T_i$ of length $3B$ bits, then we can find a prime $p_i$ near $T_i$ with $9/8B$ bits of precision. Thus, we can choose $T_i$ so that $\log T_i \propto t_i$ with precision $9/8\, B$ bits. This lets us find $p_i \approx T_i$ so that $\log p_i \propto t_i$ with precision $9/8\,B$ bits. If a subset of these primes multiplies to something close to the target value, a solution exists to the original subset sum problems. So we let $N=\Pi_i p_i$, choose $L$ and $U$ appropriately, and we have a randomized reduction from subset sum.