Why would anyone use an elliptic curve with a cofactor > 1? In cryptography, an elliptic curve is a group based on a finite field $GF(p^k)$; this group has $n$ elements on it, and we work on a prime-sized subgroup of size $q$.  We denote the value $h = n/q$ as the cofactor of the curve.

My question is: why would we ever want to consider using a curve which has a cofactor $h&gt;1$?  Or, in other words, why would we consider using an elliptic curve that had a composite number of points?  After all, the discrete log problem can be solved in $O(\sqrt{n / h})$ time; if we were to select a curve with $h&gt;1$, we are deliberately making this problem easier.

Now, if $h$ is small, we're not making it much easier; if (for example) $p^k \ge 2^{256}$ and $h \le 4$, this would still appear to be an intractible problem.  On the other hand, I don't know if we want to make the attacker's job any easier than necessary, unless we gain some other benefit from it (perhaps making some other attack harder, or gaining some computational efficiency).

So, is there any benefit for using a curve with a cofactor > 1?
Montgomery and twisted Edwards curves have even order, but the group law can be implemented using fewer multiplications than Weierstrass models.  So that is why these curves are popular and we have to live with cofactors $&gt; 1$.

There are other reasons to prefer to use prime-order elliptic curves (e.g., small subgroup attacks).  So you are right that in an ideal world one would used prime order curves.  But sometimes it is worth the trouble to use a more efficient curve model.