By Ted Bastin

An essay within the conceptual foundations of physics. Its function is to introduce what's referred to as a combinatorial strategy.

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For any source state s and any encoding rule e define e(s) =< s, e > . Prove that the resulting code is a 2-fold perfect Cartesian one of Type II with |S | = q n−1 , |E | = |M | = q n and P0 = P1 = 1/q if the encoding rules have a uniform probability distribution. Copyright 2006 by Taylor & Francis Group, LLC Chapter 4 Authentication Schemes with Arbitration We have discussed several properties of the authentication schemes with three participants: the information-theoretic bound for the successful probability of spoofing attack, the lower bound for the number of encoding rules, and the characterization of perfect schemes in Chapter 3.

There are totally q t such blocks. For each block B(at−1 , · · · , a0 ) we define an encoding rule e such that t−1 ai xi , ∀x ∈ S e(x) = i=0 Thus, we have |E | = q . We can see that |M (x)| = q for all x ∈ S . Take t distinct elements xi (1 ≤ i ≤ t) of Fq . For any r (1 ≤ r ≤ t) points mr = {(x1 , y1 ), (x2 , y2 ), · · · , (xr , yr )}, if they are in a block B(at−1 , at−2 , · · · , a0 ), then they satisfy t + at−2 xt−2 + · · · + a1 x1 + a0 = y1 at−1 xt−1 1 1 at−1 xt−1 + at−2 xt−2 + · · · + a1 x2 + a0 = y2 2 2 ·················· at−1 xt−1 r + at−2 xt−2 r + · · · + a1 xr + a0 = yr This is a system of linear equations with unknown at−1 , at−2 , · · · , a0 .

This is a well-known inequality. 4 H(X|Y ) ≤ H(X). PROOF We have H(X|Y ) − H(X) = − p(x, y) log p(x|y) + x,y = p(x, y) log x,y p(x, y) log p(x) x,y p(x)p(y) . 3 that H(X | Y ) − H(X) ≤ log ( p(x, y) x,y p(x)p(y) ) = 0. 4 means that the indeterminacy of X could decrease and could not increase when Y is given. One may find some information about X from the given Y . The decrease of indeterminacy H(X) − H(X|Y ) is the lost amount of information. 3) holds for any integer r ≥ 0. 4) is independent of mr , m, and e ∈ E (mr ∗ m).

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