Cryptography

Download An Introduction to Cryptology by Henk C. A. van Tilborg (auth.) PDF

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By Henk C. A. van Tilborg (auth.)

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This contradicts the minimality of L. D. 6. P3: If ±ld i < 1, one can decrease i,. 4). 6 a prefix code with smaller expected length would exist. This contradicts our assumption on C. P4: If (,,-I < 1" one cannot have equality in P3), because of PI). P5: Delete the last coordinate of f" and call the resulting vector f~. • f"-l>f~). 4). So C* is not a prefix code, while C was. This is only possible if f~ is a proper prefix of some codeword fi' 1 ~ i ~ n -l. Since li ~ 1" ,this proves that fi and f,.

The set {fI> f2. . • f,,} is called a code C for the source S. The idea of data compression is to use such a code that the expected value of the length of the encoded plaintext is minimal. 1) L = LPil;. i=1 over ali possible codes C for the source S. There is however an additional constraint. One has to be able to retrieve the individual messages from the concatenation of the succesive codewords. Not every code has this property. lndeed let C = {O, 01, 1O}. The sequence 010 can be made in two ways: O followed by 10 and OI followed by O.

1 Consider the f1ipping of a coin. Let Pr (heads ) = P and Pr {tails } = 1 - P, O~ P ~ l. 4). That h (1/2) = 1 is of course confirmed by the fact that one needs one bit to represent the outcome of the tossing of a fair coin. For instance 0= heads and 1 = tails. 8113 bits are needed to represent the outcome of a tossing of an unfair coin with Pr {heads} = 1/4. 8113 arbitrarily closeby. In Chapter V we sball show how to do this. The trick will be to represent the outcome of n consecutive tossings by a string of bits.

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