Mathematical Problem of Public Discussion. The purpose for the public discussion is to allow Xxxxx, A, and Bob, B, to discover what portion of their common bit sequences are identical by a public discussion without revealing what exactly these common bit sequence values are to an eavesdropper E. In this section the mathematical public discussion problem discuss mathe- matically. Also, the following declares the relationship between this problem and forward error control coding. { } Assume that N reference points, I = I1, I2, ..., IN , being selected on the integer axes and everybody knows these reference points. also assuming two random real Ra and Rb which being written on a piece of paper and being put in boxes A and B respectively. For creating Ra and Rb, we know that creator used one of the integer values of the set I and added it with a random real variable ea and eb for A and B respectively. Suppose that ea and eb are independent and have zero mean normal distribution with variance σ2. No body else except the creator of Ra and Rb knows these values. Ra = Ip + ea Rb = Ip + eb where Ip ∈ I. Xxxxx opens the box A and Bob opens the box B. They want to agree on an integer number, Ip, together in front of Xxx. Also they want to provide security of their agreement. In secrecy communication this discussion for agreement on a shared secret has been nominated public discussion (i.e. the discussion that is public and obvious to Xxx but just help the legitimate users to agree and does not give any useful information to Xxx about Ip (what A and B want to agree on). Suppose that Bob finds the nearest reference integer in I to its own number, Rb, and announces the difference between Rb and its selected reference, Ib, to Xxxxx. Xxxxx looks for an Ia ∈ I that minimizes Ia + Rb − Ib − Ra. Ia + Rb − Ib − Ra = (Ia − Ib)+ e (2.1) e = eb − ea (2.2)
