Physical Unclonable Functions (PUFs) are promising primitives for the lightweight authentication of an integrated circuit (IC). Indeed, by extracting an identifier from random process variations, they allow each instance of a design to be uniquely identified. However, the extracted identifiers are not stable enough to be used as is, and hence need to be corrected first. This is currently achieved using error-correcting codes in secure sketches, that generate helper data through a one-time procedure. As an alternative, we propose key reconciliation protocols. This interactive method, originating from quantum key distribution, allows two entities to correct errors in their respective correlated keys by discussing over a public channel. We believe that this can also be used by a device and a remote server to agree on two different responses to the same challenge from the same PUF obtained at different times. This approach has the advantage of requiring very few logic resources on the device side. The information leakage caused by the key reconciliation process is limited and easily computable. Results of implementation on FPGA targets are presented, showing that it is the most lightweight error-correction module to date.