The One-Message Key Rate Clause Samples

The One-Message Key Rate. This previous discussions give rise to the following definition, which we can trace back to [AC93].1 3.1 ( One-message key rate). Let PXYZ be any probability distri- bution over X × Y × Z. The one-message key rate S→(X; Y|Z) is S→(X; Y|Z) := sup PUV|X H(U|ZV) — H(U|YV), where the supremum is over all conditional distributions PUV X with finite al- phabets for U and V. In our previous example we obtained U and V in a deterministic way from X. However, in general this is not possible (we will see an example in Chapter 4). | → | | — | It will follow from Theorem 3.3 that there always exists a distribu- tion PUV X with S (X; Y Z) = H(U ZV) H(U YV), i.e., the supremum can always be achieved and is finite. The naming of S (X; Y Z) as one-message key rate originates from Theorem 3.13, which states that for any rate R < S (X; Y Z) it is possi- ble to obtain nR key bits from n random variables, as long as n is large enough; as well as from Theorem 3.18, which states that no rate R > S (X; Y Z) can be achieved by any protocol. Finally, we note that the expression we maximize over in Definition 3.1 can be equivalently written as H(U|ZV)—H(U|YV) = H(UZV) — H(ZV) — H(UYV) + H(YV) = H(Z|UV) — H(Y|UV) — (H(Z|V) — H(Y|V)). (3.1)
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