Kolmogorov complexity and cryptography

Abstract

We consider (in the framework of algorithmic information theory) questions of the following type: construct a message that contains different amounts of information for recipients that have (or do not have) certain a priori information. Assume, for example, that a recipient knows some string a and we want to send him some information that allows him to reconstruct some string b (using a). On the other hand, this information alone should not allow the eavesdropper (who does not know a) to reconstruct b. This is indeed possible (if the strings a and b are not too simple). Then we consider more complicated versions of this question. What if the eavesdropper knows some string c? How long should our message be? We provide some conditions that guarantee the existence of a polynomial-size message; we show then that without these conditions this is not always possible.