Download Algorithmic Information Theory: Mathematics of Digital by Peter Seibt PDF

By Peter Seibt

Algorithmic info idea treats the math of many very important parts in electronic details processing. it's been written as a read-and-learn booklet on concrete arithmetic, for lecturers, scholars and practitioners in digital engineering, machine technology and arithmetic. The presentation is dense, and the examples and workouts are various. it's in accordance with lectures on info know-how (Data Compaction, Cryptography, Polynomial Coding) for engineers.

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Example text

2 The Cipher DES in Detail 1. The initial permutation IP : It is described by the following table: 58 50 60 52 62 54 64 56 57 49 59 51 61 53 63 55 You have to read line by line: 42 44 46 48 41 43 45 47 IP 34 36 38 40 33 35 37 39 26 28 30 32 25 27 29 31 18 20 22 24 17 19 21 23 10 2 12 4 14 6 16 8 9 1 11 3 13 5 15 7 IP (t1 t2 · · · t63 t64 ) = t58 t50 · · · t15 t7 . 54 2 Cryptography IP will shuffle, in a very regular manner, the bit-positions of the eight 8-bit bytes which constitute the block T of 64 bits: every new byte contains precisely one single bit of every old byte.

42, I(b) = 3, I(c) = I(d) = 4. The Shannon code: a b c d • − − − − − − − − − − −− • −− • − • − • 3 7 15 1 0 4 8 16 a −→ 0 b −→ 110 c −→ 1110 d −→ 1111 Let us choose a source word in conformity with the statistics: daaabaaacaaabaaa. The associated Shannon code word is 11110001100001110000110000 and has 26 bits. 04+6+8 = 20. So, the arithmetic code word of daaabaaacaaabaaa is shorter than the (concatenated) Shannon code word. This is a general fact: whenever the probabilities are not powers of 12 , arithmetic coding is better than any block coding (of fixed block length).

Find the arithmetic code word of aabaacaa. (4) Write down the general version of the recursive algorithm for arithmetic coding. Recall : we have to do with a memoryless source producing N letters a0 , a1 , . . , aN −1 , according to the probability distribution p = (p0 , p1 , . . , pN −1 ), and p0 ≥ p1 ≥ · · · ≥ pN −1 > 0. (5) The situation as in exercise (4). Suppose that all probabilities are powers of 12 : pj = 2−lj , 0 ≤ j ≤ N − 1. Show that in this case the arithmetic code word of a source word s1 s2 · · · sn is equal to the Shannon code word (obtained by simple concatenation of the code words for s1 , s2 , .

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