Download e-book for kindle: An Introduction to Cryptology by Henk C. A. van Tilborg (auth.)

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

ISBN-10: 1461289556

ISBN-13: 9781461289555

ISBN-10: 1461316936

ISBN-13: 9781461316930

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O, 1, O, ... , O) oflength 12 , with alon coordinate n, has length li' It remains to show that no fu can be the prefix of a codeword f •. Suppose the contrary. Clearly 1" 1•• So 1" < 1. and thus u < v. -1 L -/. - L -/. = L -/ ~ -I-o ;=1 2 J ;=1 2 J o A contradiction! 3 resp. 6 are the same. D. codes is equal to the smallest expected value of the length among alI prefix codes! D. codes. D. code with codewords fi of length li for the messages mi that occur with probability Pi, 1::;;; i ~ n. ). Pi· In--/.

We proceed by induction on n. Let C be a Huffman code for a source S with n symbols. Let the codewords fi of C have length li, 1 ~ i ~ n, and let L be the expected length of a codeword in C. D. D. codes for S. 12 both codes, C and C* , satisfy properties PI-PS. -l and f... in C differ only in their last coordinate, as do f~-l and f~ in C* . Now apply one step of the reduction process to C and C*. One obtains a Huffman code D and a prefix code D* , with expected lengths M resp. M* . By the induction hypothesis M ~ M* .

843 bits per tossing. etc. gives better approximations of h (114). There is however the problem that the receiver of a long string of zeros and OIles should be able to determine the outcomes of the tossings in a unique way. One can easily verify that any sequence made up from the subsequences 111, IlO, 10 and O, can only in one way be broken up into these subsequences. We come back to this problem in Chapter V. 1 The function h (p), 0'5, p '5, 1. 70 bits per letter, by coding sufficiently long strings of letters into binary strings.

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


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