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I am working on the product of two perfect crypto-systems and I need to prove that the product is secure.

$$a -- [\text{system}\ 1] -- b -- [\text{system}\ 2] -- c$$

How can I prove that $H(a) = H(a|c)$ knowing that $H(a|b) = H(a)$ and $H(b|c) = H(b)$?

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You need something on system 1 and system 2 being independent, because if you compose (e.g.) 2 one-time pads, the result is not so secure.... –  Henno Brandsma Feb 10 '12 at 21:57
Isn't this a markov chain? $p(c|a,b) = p(c|b)$ by markovity, and $p(c|b) = p(c)$ by independence of $b$ and $c$. $p(c|a,b) = p(c)$ also implies $a$ and $c$ are independent –  VSJ Feb 10 '12 at 22:09
The two systems are independent I think. –  Romain Pignard Feb 11 '12 at 1:08

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