Kolmogorov complexity and cryptography

Abstract: This paper contains some results of An.A.Muchnik (1958-2007) reported in his talks at the Kolmogorov seminar (Moscow State Lomonosov University, Math. Department, Logic and Algorithms theory division, March 11, 2003 and April 8, 2003) but not published at that time. These results were stated (without proofs) in the joint talk of Andrej Muchnik and Alexei Semenov at Dagstuhl Seminar 03181, 27.04.2003-03.05.2003. This text was prepared by Alexey Chernov and Alexander Shen in 2008-2009. 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 the recipient knows some string $a$, and we want to send her some information that allows her 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 $b$. It 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 be our message? 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.