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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Proof' For n = 2 the statement is obvious. 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.

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 }:. i=1 ••• ::; '" and " Pi = 1. ::; }:. 6. ) + 1. ) codes for a source S with probabil- ity distribution 2 bas a value L satisfying H(2)::; L < H(2) + 1. ) by one bit or more. 8 to N -tuples of source symbols, one gets in the same way an expected length L IN) per N -gram, satisfying N • H(2)::; LIN) < N • H(2) + 1.

Otherwise the output sequence {Si Ci = O then the corresponding switch in We shall always assume that Co = 1, because };"o is just a delayed version of a sequence, generated by a LFSR with its C o equal to 1. As a consequence any state of the LFSR not only has a unique successor state, as is natural, but also has a unique predecessor. Later we shall prove this property in a more general situation. 2 With n ... output time O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 C2 = C3 = O and starting state (1,0,0,0) one gets the following ~ 1 O O O 1 O O 1 1 O 1 O 1 1 1 1 O O O 1 O O O O 1 O O 1 1 O 1 O O 1 O 1 1 1 1 O O O O 1 O O 1 1 1 1 O O 1 1 1 1 1 1 1 1 1 O O O , etc ..

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