Do encrypted Time Machine backups work across a network?
According to information from this page, the answer is no, it will not support encrypted Time Machine backups to network-attached storage. They're specifically mentioning Time Capsules and Airport Extremes. The issue Paul Eccles mentioned looks to be related to whether you can backup to NAS with Time Machine at all without regard to encryption status.

The link referenced also seems to indicate you can't use Time Machine encryption on direct-attached storage, but that may be because you can use Disk Utility to encrypted a direct-attached volume instead. Commenter Fracai calls out the classic standby of creating an encrypted disk image, storing that on network-attached storage, and mounting that for your Time Machine backups.

I'll also point out the discussion on that page (which is slightly inaccurate, but mostly on) wherein commenter TrumpetPower! cautions encrypting backups. This is a point worth repeating here. I'll rephrase (more precisely and accurately) what TrumpetPower! wrote:


The purpose of a backup is to preserve availability of data.
The purpose of encryption is to prevent availability of data without the key used to encrypt the data.


If you encrypt a backup, you add complexity to a process designed to simplify and preserve access to your information. With complexity comes increased probability of failure. Thus, the chance that you lose access to your backed-up data goes up. Do yourself a favor and make sure you understand when and why you might wish to encrypt your backups and think hard about how you'll ensure you have access to your encryption key when it counts.