  Privacy amplification (PA) is an indispensable component in classical and
quantum cryptography. Error correction (EC) and data compression (DC)
algorithms are also indispensable in classical and quantum information theory.
We here study these three algorithms (PA, EC, and DC) in the presence of
quantum side information, and show that they all become equivalent in the
one-shot scenario. As an application of this equivalence, we take previously
known security bounds of PA, and translate them into coding theorems for EC and
DC which have not been obtained previously. Further, we apply these results to
simplify and improve our previous result that the two prevalent approaches to
the security proof of quantum key distribution (QKD) are equivalent. We also
propose a new method to simplify the security proof of QKD.
