There are essentially two ways to find out what the time is: ask an authoritative source and trust the answer, or ask several more or less unreliable sources and see what they agree on. NTP is based on the latter principle, but since the protocol isn't secured, a client also has to trust the network not to turn NTP responses into lies.
NTP's lack of security causes a bootstrapping problem. Many security protocols rely on accurate time to avoid replay attacks. So nearly the first thing a networked device needs to do on startup is get the time, so that it can then properly verify what it gets from the network - DNSSEC signatures, TLS certificates, software updates, etc. This is particularly challenging for cost-constrained devices that do not have a battery-backed real time clock and so start up in 1970.
When I say NTP isn't secured, I mean that the protocol has security features but they have not been deployed. I have tried to understand NTP security, but I have not found a description of how to configure it for the bootstrap case. What I want is for a minimally configured client to be able to communicate with some time servers and get responses with reasonable authenticity and integrity. Extra bonus points for a clear description of which of NTP's half dozen identity verification schemes is useful for what, and which ones are incompatible with NATs and rely on the client knowing its external IP address.
In the absence of usable security from NTP, Jacob Appelbaum of the Tor project has written a program called tlsdate. In TLS, the ClientHello and ServerHello messages include a random nonce which includes a Unix time_t value as a prefix. So you can use any TLS server as a secure replacement for the old port 37 time service.
Unlike NTP, tlsdate gets time from a single trusted source. It would be much better if it were able to consult multiple servers for their opinions of the time: it would be more robust if a server is down or has the wrong time, and it would be more secure in case a server is compromised. There is also the possibility of using multiple samples spread over a second or two to obtain a more accurate time than the one second resolution of TLS's gmt_unix_time field.
The essential idea is to find a quorum of servers that agree on the time. An adversary or a technical failure would have to break at least that many servers for you to get the wrong time.
In statistical terms, you take a number of samples and find the mode, the most common time value, and keep taking samples until the frequency at the mode is greater than the quorum.
But even though time values are discrete, the high school approach to finding the mode isn't going to work because in many casees we won't be able to take all the necessary samples close enough together in time. So it is better to measure the time offset between a server and the client at each sample, and treat these as a continuous distribution.
The key technique is kernel density estimation. The mode is the point of peak density in the distribution estimated from the samples. The kernel is a function that is used to spread out each sample; the estimated distribution comes from summing the spread-out samples.
NTP's other algorithms are based on lengthy observations of the network conditions between the client and its servers, whereas we are more concerned with getting a quick result from many servers. So perhaps we can use a simpler, more well-known algorithm to find the mode. It looks like the mean shift algorithm is a good candidate.
For the mean shift algorithm to work well, I think it makes sense to use a smooth kernel such as the Gaussian. (I like exponentials.) The bandwidth of the kernel should probably be one second (the precision of the timestamp) plus the round trip time.
Now it's time to break out the editor and write some code... I think I'll call it "temporum" because that rhymes with "quorum" and it means "of times" (plural). Get temporum from my git server.