Security Proofs for Information Protection Systems

Recently discovered procedures use a random input, rather than a cryptographic key, to turn a piece s of information into n + 1 pieces of information in such a way that s can be recovered from any b + 1 of them but that it is hard, or perhaps impossible in a sense which must be precisely defined, to recover s from any b of them. Thus, for example, one might have 15 pieces of information such that any 9 of them suffice to reconstitute s, but euch that no 8 of them give any hint as to what s is. The various authors of euch procedures have called them key safeguarding schemes, threshold schemes, secret sharing, and key sharing. None of these names captures the idea, which we will denote by information protection system (IPS). Our purpose is to put a rigorous foundation under the intuitive security arguments these papers adduce. In the proceed we will produce a distinctive style of proof of security, a rigorous argument involving product measures as a conceptual basis for justifying intuitively plausible probabilistic statement of the sort C. E. Shannon used to describe the security of the one-time pad.