By Bohlin T.

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2 Preliminaries 19 Message m’ m1 ,1 h1 m2 ,1 h2 m|m'|/|t| ,1 hlog(|m’|/|t|) Split key h m1 m2 m3 mn r1 rn mj ,k= hk-1( m2j-1 ,k-1)||hk-1( m2j ,k–1) if mj ,k does not exist mj ,k=0 t’= lb|t’|( hlog(|m’|/|t|)( m1 ,log(|m’|/|t|)) ) r2 r3 ti = h(mi) Å ri t’ t1 t2 t3 tn Fig. 3 Entropy Entropy is a key concept in the field of information theory, which is a branch of applied mathematics concerned with the process to quantifying information. In the following a short part is presented, which we need in later chapters.

36 M. Pivk With the fact that a ideal reconciliation protocol depends on a open question in complexity, which is unlikely to be true, we have to find another solution. In Sect. 1 (universal hashing) we had a similar problem. There we define a str ongly univer sal2 class of hash functions, where the size of this class becomes too large and unpractical. But just a small change of the probability on the theoretical bound makes these classes useful. Here again if we do not demand optimality and allow the reconciliation protocol to transmit a small amount of leaked information above the theoretical bound the protocol becomes efficient and useful.

The next theorem is a direct consequence of the noiseless coding theorem. 1 (∀ p ≤ 12 ) (∀reconciliation protocol R p ) If there exists 0 ≤ ε ≤ 1 such that R p = [S, Q] is ε-robust then lim n→∞ I E (S|Q) ≥ 1, nh( p) 3 Quantum Key Distribution 35 where n is the length of the transmitted string. In other words, Eve’s information of S is greater than or equal to the information of A given B. Eve cannot have less than this information if the reconciliation protocol R p is successful. But if these two pieces of information are equal the protocol is optimal.