## Algorithmic Information Theory: Mathematics of Digital Information ProcessingShall we be destined to the days of eternity, on holy-days,as well as working days, to be shewing the RELICKS OF LEARNING, as monks do the relicks of their saints – without working one – one single miracle with them? Laurence Sterne, Tristram Shandy This book deals with information processing; so it is far from being a book on information theory (which would be built on description and estimation). The reader will be shown the horse, but not the saddle. At any rate, at the very beginning, there was a series of lectures on “Information theory, through the looking-glass of an algebraist”, and, as years went on, a steady process of teaching and learning made the material evolve into the present form. There still remains an algebraic main theme: algorithms intertwining polynomial algebra and matrix algebra, in the shelter of signal theory. A solid knowledge of elementary arithmetic and Linear Algebra will be the key to a thorough understanding of all the algorithms working in the various bit-stream landscapes we shall encounter. This priority of algebra will be the thesis that we shall defend. More concretely: We shall treat, in ?ve chapters of increasing di?culty, ?ve sensibly di?erent subjects in Discrete Mathem- ics. The?rsttwochaptersondatacompaction(losslessdatacompression)and cryptography are on an undergraduate level – the most di?cult mathematical prerequisite will be a sound understanding of quotient rings, especially of- nite ?elds (mostly in characteristic 2). |

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### Contents

5 | |

10 | |

113 Arithmetic Coding | 32 |

The Example LZW 121 LZW Coding | 43 |

122 The LZW Decoder | 45 |

Cryptography | 49 |

211 The DES Scheme | 50 |

212 The Cipher DES in Detail | 53 |

32 Trigonometric Interpolation | 190 |

321 Trigonometric Polynomials | 191 |

331 Fourier Series | 198 |

332 The WhittakerShannon Theorem for Elementary Periodic Functions | 203 |

A Sketch | 209 |

334 The Sampling Theorem | 214 |

Error Control Codes 41 The ReedSolomon Codes 411 Preliminaries Polynomial Codes | 220 |

412 ReedSolomon Codes | 225 |

The Cipher Rijndael 221 Some Elementary Arithmetic | 60 |

222 Speciﬁcation of Rijndael | 77 |

223 The Key Schedule | 86 |

224 Decryption with Rijndael | 92 |

231 Encryption and Decryption via Exponentiation | 93 |

232 The Cryptosystem RSA | 97 |

241 Message Digests via SHA1 | 101 |

Digital Signature Algorithm | 112 |

243 Auxiliary Algorithms for DSA | 116 |

244 The Signature Algorithm rDSA | 122 |

245 ECDSA Elliptic Curve Digital Signatures | 125 |

Information Theory and Signal Theory Sampling and Reconstruction | 171 |

311 Basic Properties | 172 |

312 The Fast Fourier Transform Algorithm | 183 |

421 Encoding Digital Filtering in Binary Arithmetic | 239 |

The Viterbi Method | 253 |

Data Reduction Lossy Compression | 267 |

51 DFT Passband Filtering and Digital Filtering | 268 |

52 The Discrete Cosine Transform | 274 |

521 Functional Description of the DCT | 275 |

522 The 2D DCT | 293 |

523 The KarhunenLoeve Transform and the DCT | 305 |

531 Two Channel Filter Banks | 314 |

532 The Discrete Wavelet Transform | 372 |

435 | |

438 | |

### Other editions - View all

Algorithmic Information Theory: Mathematics of Digital Information Processing Peter Seibt No preview available - 2010 |

Algorithmic Information Theory: Mathematics of Digital Information Processing Peter Seibt No preview available - 2006 |

### Common terms and phrases

algorithm arithmetic coding bbbbbbbb bbbbbbbb bbbbbbbb binary notation binary polynomial binary words biorthogonality bits ciphertext CM discriminant code bitstream code word coeﬃcients Compute Consider convolutional code decoding decryption deﬁned deﬁnition diﬀerent Discrete Fourier Transform Discrete Wavelet Transform DWT 5/3 spline ECDSA eigenvalues elliptic curve encoder encryption equation Example Exercises ﬁeld ﬁlter filter bank ﬁnd ﬁnite ﬁrst ﬁxed formal Hence high-pass Huﬀman code impulse responses information bitstream integer interleaved interval inverse ip(t JPEG linear low-pass mathematical matrix memoryless source message digest modulo multi-resolution analysis multiplication non-zero Note obtain orthogonal orthonormal basis perfect reconstruction plaintext polynomial of degree precisely prime number probability distribution quadratic quantization Recall Reed–Solomon codes round key S-box sample scaling function scheme sequence Show signal theory signature situation step symmetric symmetric matrix synthesis theorem transmission error trigonometric polynomial values vector words of length zero