Common randomness in information theory and cryptography. I. Secret sharing

One-Way Secret-Key Agreement and Applications to Circuit Polarization and Immunization of Public-Key Encryption

“…For one-way communication, it is already implied by a result in [3] and has later been shown in [1] that the secret-key rate S → (X; Y |Z) is given by the supremum of H(U |ZV ) − H(U |Y V ), taken over all possible random variables U and V obtained from X. 1 However, as this is a purely information-theoretic result, it does not directly imply that there exists an efficient key-agreement protocol.…”

“…As mentioned in the introduction, Theorem 1 has already been known to hold for general (not necessarily efficient) protocols [3,1].…”

“…1 However, as this is a purely information-theoretic result, it does not directly imply that there exists an efficient key-agreement protocol.…”

“…He considered a setting where Alice, Bob, and Eve hold many independent realizations of correlated random variables X, Y , and Z, respectively, with joint probability distribution P XY Z . The (two-way) secret-key rate S(X; Y |Z), i.e., the rate at which Alice and Bob can generate secret-key bits per realization of (X, Y, Z), has further been studied in [1] and later in [12], where the intrinsic information I(X; Y ↓Z) is defined and shown to be an upper bound on S(X; Y |Z), which, however, is not tight [13].…”

“…Maurer [11] (cf. also [1]) proposed to consider a setting where a satellite broadcasts uniform random bits with low signal intensity, such that Alice, Bob, and also Eve cannot receive them perfectly. It has been shown that, in both settings, Alice and Bob can indeed generate an information-theoretically secure key and thus communicate secretly.…”