By Henk C. A. van Tilborg (auth.)
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Additional resources for An Introduction to Cryptology
A 1 at the end. The other symbol keeps its original 52 AN INTRODUcrION TO CRYPTOLOGY encoding. Repeat this splitting up process until the original alphabet is back and ali the symbols are encoded. The code obtained in this way is called a Huffman code. 11: n = 6. 10. Proof: The proof is straightforward with an induction argument. 13 Pl~P2~ Let S be a source with symbols mi, 1 ~ i ~ n, with resp. probabilities ... ~p". Let C be a Huffman code for this source with lengths li, l~i~n, and expected length L.
D. codes. D. code with codewords fi of length li for the messages mi that occur with probability Pi, 1::;;; i ~ n. ). Pi· In--/. -/ In2 i=12 i In2 i=1 Pi. 2 • o 1)::; O. 8 Let S be a source generating symbols mi with probabilities Pi, 1::; i ::; 11. Then a prefix code C exists for this source with expected length L < H (p) + 1. Proof· Without loss of generality p 1 ~ P 2 ~ ... ~ p". Define li by POg2 11 Pi 1, 1::; i ::; n. Here rx 1denotes the smallest integer greater than or equal to x. Clearly 'I::; 1 2 ::; " 1/2/i }:.
Continuing in this way one gets AN INTRODUCTION ro CRYPTOLQGY 40 J (1/2") = k, k ~ O. ), I!. = (PhP2> ... = x}) is called the entropy of X and will be denoted by H(X) or ,p,,). ) = H (X) = E (J (Prx (X = x) " " log2Pi. 3) i=1 One can give the following interpretations to H(X): the expected amount of information from a realization of X , our uncertainty about X, the expected number of bits needed to descibe an outcome of X. With this interpretation in mind one expects the entropy function H(X) to have the following properties: PI: H(PhP2, ...
An Introduction to Cryptology by Henk C. A. van Tilborg (auth.)