How does Apple's Touch ID fingerprint sensor work?
There are a few different fingerprint-sensor technologies out there, with optical and capacitance readers being the most common. Optical readers take a picture of your fingerprint with a digital camera. Apple chose a capacitance reader, which is far more interesting.
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A capacitance fingerprint reader leverages a handy property of your skin: The outer layer of your skin (your dermis), where your fingerprint is, is non-conductive, while the subdermal layer behind it is conductive. When you touch the iPhone's fingerprint sensor, it measures the minuscule differences in conductivity caused by the raised parts of your fingerprint, and it uses those measurements to form an image..
Apple embedded this sensor in the Home button, and added a ring to turn it on and help reduce signal errors. I suspect that the ring also adds a little current to your finger to help boost and clean the signal.
It appears to be a great design. Most other portable readers I've used in the past were optical, which is easier to fool (sometimes a good photocopy will work), easier to break, and more prone to error (thanks to smudged glass and other factors).
Does my iPhone store my fingerprint?
Apple says no, and here is what I think is going on. Typically, your fingerprint is scanned and run through a mathematical algorithm that creates a fingerprint template. This template is a representation of part of your fingerprint; it isn't a stored image.
Better yet, most advanced systems run this template through a cryptographic hashing algorithm, as they do for passcodes, and store that result. To add even more security, during hashing it is combined with a unique or random number to make recovery even harder. Since your iPhone already does this with your passcodes (using a special device ID embedded in your hardware), I suspect Apple uses the same process for your fingerprint template.
Every time you scan your fingerprint, the phone runs through the same algorithmic process and the result is compared with the stored hash. Not only is your actual fingerprint not stored, but it's likely really hard or impossible to recover even if the NSA gets your phone.
I'm assuming a few things here, but they're educated assumptions based on how Apple manages passcodes today.
Is a fingerprint more secure than a passcode?
Fingerprints are both more and less secure than passcodes. A fingerprint is more secure since it is effectively impossible to guess. It can be less secure since, if someone steals it once, they steal it for life.