Henk C. A. van Tilborg (auth.)'s An Introduction to Cryptology PDF

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

ISBN-10: 1461289556

ISBN-13: 9781461289555

ISBN-10: 1461316936

ISBN-13: 9781461316930

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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--/. -/ 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.

D. code for this source with codewords ... , P5) two of the codewords of length 1,. differ only in their last coordinate. Proof: PI: Suppose that Pu > py and iu > Iy. Make the code C* from C by interchanging fu and fy. Then the expected length L * of C* satisfies L * = L + Pu (ly - lu) + py (lu - Iy) = L + (p,. D. code. 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.

Iog2 (1-p)= l-H(P). So the receiver gets 1 - H (p) bits of infonnation about X per received symboI Y. 1. §6]. For p = 1/2 the receiver gets DO infonnation about the transmitted symboIs. as is to be expected. Let us now retum to the conventional cryptosystem as explained in Chapter I. M 1• . . • M,,_I) denote the plaintext. • C,_I) denote the ciphertext. So C' = EK (M"). C") = o. 13) Of course the user of the cryptosystem is interested in how much infonnation C' contains about M". 91 (M";C') '?

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


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