Apple has embedded a fingerprint sensor into the iPhone 5s's Home Button as a way to bypass the lock screen's passcode. This is Touch ID.
All fingerprint information is encrypted and stored in the A7 chip. Touch ID it does stores "mathematical representation" of fingerprints.
The iPhone 5s has an advanced security architecture called the Secure Enclave - within the A7 chip that protects and verifies fingerprint matches. Apple said the Secure Enclave is "walled off from the rest of A7 and as well as the rest of iOS", meaning only Touch ID has access to fingerprint data.
TouchID has 4 header files:
PSBiometricIdentity.h SBUIBiometricEventMonitor.h SBUIBiometricEventObserver.h BiometricKitDelegate.h
Which can all be found at: developer.limneos.net.
The PSBiometricIdentity.h is responsible for the settings of TouchID which can be found in settings >General > TouchId & Passcode > Touch ID. The 2 SpringBoardUIServices headers are responsible for scanning and detecting the finger on the lock screen; they use the delegate methods to for example match the finger like in the -(void)matchResult:(id)arg1; method.
If you log the argument of that method you will get something around the lines of: -[<SBUIBiometricEventMonitor: 0x17867c3c0> matchResult:<BiometricKitIdentity: 0x1782562f0>] . That <BiometricKitIdentity: 0x1782562f0> is the name of the finger that was just scanned and verified. Now we know from Apple that Touch ID is stored on the A7 Processor chip inside the 5s in a secure enclave. What do we not know? How much storage is their in this secure enclave? Is it variable?...