Chinese Remainder Theorem Applications In Computing Coding Cryptography
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Chinese Remainder Theorem: Applications In Computing, Coding, Cryptography
Chinese Remainder Theorem, CRT, is one of the jewels of mathematics. It is a perfect combination of beauty and utility or, in the words of Horace, omne tulit punctum qui miscuit utile dulci. Known already for ages, CRT continues to present itself in new contexts and open vistas for new types of applications. So far, its usefulness has been obvious within the realm of “three C's”. Computing was its original field of application, and continues to be important as regards various aspects of algorithmics and modular computations. Theory of codes and cryptography are two more recent fields of application.This book tells about CRT, its background and philosophy, history, generalizations and, most importantly, its applications. The book is self-contained. This means that no factual knowledge is assumed on the part of the reader. We even provide brief tutorials on relevant subjects, algebra and information theory. However, some mathematical maturity is surely a prerequisite, as our presentation is at an advanced undergraduate or beginning graduate level. We have tried to make the exposition innovative, many of the individual results being new. We will return to this matter, as well as to the interdependence of the various parts of the book, at the end of the Introduction.A special course about CRT can be based on the book. The individual chapters are largely independent and, consequently, the book can be used as supplementary material for courses in algorithmics, coding theory, cryptography or theory of computing. Of course, the book is also a reference for matters dealing with CRT.
Chinese Remainder Theorem
Author: Cunsheng Ding
language: en
Publisher: World Scientific Publishing Company Incorporated
Release Date: 1996
1. Introduction and philosophy. 1.1. A historical overview. 1.2. Pars pro toto. 1.3. Chinese remainder theorem: a first formulation. 1.4. CRT in the hands of old mathematicians. 1.5. CRT in applications: the three C's -- 2. Chinese remainder algorithm. 2.1. Historical development. 2.2. Chinese remainder algorithms. 2.3. Chinese remainder theorem. 2.4. A generalized CRA. 2.5. Another generalized CRT -- 3. In modular computations. 3.1. Modular computation based on CRA. 3.2. A modular approach to multiplication. 3.3. Computing exact polynomial resultants. 3.4. Other applications in symbolic computations. 3.5. CRA and homomorphic image computing. 3.6. Information and CRT -- 4. In algorithmics. 4.1. Divide-and-conquer techniques. 4.2. Polynomial interpolation over fields. 4.3. Polynomial interpolation over Z/(m). 4.4. Shift-register synthesis over Z/(m). 4.5. Common primitive roots. 4.6. From one- to multi-dimension. 4.7. A modular algorithm for cyclic convolution. 4.8. A fast algorithm for cyclic convolution. 4.9. Fast fourier transform and CRT -- 5. In bridging computations. 5.1. A main bridge. 5.2. Solving equations over Z/(m). 5.3. Number of roots of equations over Z/(m). 5.4. Computing fixed points. 5.5. Bridging divisions of polynomials. 5.6. Permutation polynomials of Z/(m) -- 6. In coding theory. 6.1. Basics of block codes. 6.2. Redundant residue codes. 6.3 Reed-Solomon codes. 6.4. Redundant residue codes of degree 2. 6.5. Bossen-Yau codes. 6.6. Generalized redundant residue codes. 6.7. Restricted GRR codes. 6.8. A Class of arithmetic residue codes -- 7. In cryptography. 7.1. Secret sharing and CRT. 7.2. Secret sharing and codes. 7.3. CRT and stream ciphering. 7.4. CRA and knapsack problems. 7.5. Public-key systems via CRT
Current Problems in Applied Mathematics and Computer Science and Systems
This book is based on the best papers accepted for presentation during the International Conference on Actual Problems of Applied Mathematics and Computer Systems (APAMCS-2022), Russia. The book includes research materials on modern mathematical problems, solutions in the field of scientific computing, data analysis and modular computing. The scope of numerical methods in scientific computing presents original research, including mathematical models and software implementations, related to the following topics: numerical methods in scientific computing; solving optimization problems; methods for approximating functions, etc. The studies in data analysis and modular computing include contributions in the field of deep learning, neural networks, mathematical statistics, machine learning methods, residue number system and artificial intelligence. Finally, the book gives insights into the fundamental problems in mathematics education. The book intends for readership specializing in the field of scientific computing, parallel computing, computer technology, machine learning, information security and mathematical education.