Final answer:
In Quantum Cryptography, if Alice and Bob find discrepancies in their shared extensions (bits), it indicates an error or eavesdropping. They compare a portion of their keys to check for eavesdroppers and discard any compromised bits, using privacy amplification to secure their final key.
Step-by-step explanation:
In Quantum Cryptography, when Alice and Bob have different values for the extensions, it typically indicates an error or potential eavesdropping on the quantum channel. Quantum Cryptography employs quantum key distribution (QKD) which allows two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages.
In a typical quantum key distribution process, Alice and Bob each generate a random sequence of qubits, which are quantum bits. They choose random polarization bases for these qubits and send them to each other. After the transmission, they compare a portion of their chosen bases. If the bases match, they should have identical bits in those positions, known as 'sifted keys'. However, if their bases do not match, the values are discarded. If there are differences in the sifted keys where the bases did match, this could signal interference or eavesdropping. Such discrepancies are expected up to a certain threshold due to the inherent errors in quantum transmissions, but if the error rate goes beyond this threshold, it could compromise the security of the key exchange.
Because a fundamental property of quantum mechanics is that the act of measuring a quantum system disturbs the system, an eavesdropper trying to intercept the qubits will unavoidably introduce errors that Alice and Bob can detect. They then discard the affected bits and use techniques like privacy amplification to adjust their keys, thus ensuring the security of their communication.