Common use of P P Clause in Contracts

P P. Proof. The fact that S (PXY Z ) = 0 when either E < B or E < A follows from Theorem 5 because PXY Z is either X-simulatable or Y -simulatable by ▇▇▇. The fact that S (PXY Z ) = S(PXY Z) when E > B and E > A can be proved as follows. A suboptimal protocol based on the authentication method of Theorem 7 can be used to generate a relatively small t-bit secret key K, using O(t) bits of the random string. This key can then be used, similar to a bootstrapping process, for instance based on the protocols of [10], to authenticate the messages exchanged in an optimal passive-adversary protocol achieving S(PXY Z). The size of K must only be logarithmic in the maximal size of a message exchanged in [10] and linear in the number of rounds of . No matter what amount of secret key must be generated by , this can be achieved by using messages of size proportional to the key size in a constant number of rounds. Therefore, the ratio of size of K and the size of the generated key vanishes asymptotically. min[h( AE); h( BE)] h( AB) S(PXY Z) 1 h( AB): It was recently proved that S(PXY Z) > 0 unless E = 0 [17], even when both E < B and E < A, i.e., even when the above lower bound vanishes (or is negative).