There Are Physics And Chemistry Of The Real And Rational Approximation In The Attack On RSA Low Decryption Exponent Of Applications

The popularization of computers plays a huge role for promoting all aspects of the social life. We encounter many new problems that need to be solved as computers are widely used in various industries and the results from studying theses problems result in the emerging of various cross-disciplinary. Some problems in information science deadly need to be studied and solved from information science and mathematics. In this dissertation, we explore a representation of basic numbers so that we can better understand them and better use them in information science. Real numbers can be generally divided into rational numbers and irrational numbers, and these numbers can be approximately represented by finite decimal. In this dissertation, we first prove that it is feasible to approximately represent a real number by a finite decimal. This representation is easy to be accepted and to be understood, but it cannot be well applied in real life. To overcome this problem, we deeply study the b-ary digit real number and introduce the concept of continued fraction that is widely used in number theory. By the special properties of the continued fraction, we can know that any irrational number or rational number can be unique represented as a continued fraction. This dissertation discusses the properties and theorems related to continued fraction in detail, and verify the continued fraction algorithm, the relation between continued fraction and its asymptotic fraction through examples. Furthermore, we make error analysis of the asymptotic fraction of some commonly used real numbers. We analyze the application of continued fractions on the rational approximation, and verify the application of rational approximation to the Wiener’s low decryption exponent attack with.In this dissertation, we use continued fraction to represent a real number, that is, we use the properties of continued fraction to analyze the asymptotic fraction of a real number. We verify Wiener’s small decryption exponent attack using the asymptotic fraction of real numbers. The works of this dissertation is as follows:1、By the properties, the concept continued fraction and the properties of optimal asymptotic fraction, we analyzed the algorithm of converting finite decimals or infinite recurring decimals into fractions. Finally, we analyzed the error between some common some real constants and their asymptotic fraction2、Using the rational approximations of continued fraction, we analyzed the security of RSA public-key cryptosystem. Introducing the concept of Wiener’s low decryption exponent attack on RSA cryptosystem, we discussed how to apply the continued fraction to the low decryption exponent attack in detail. We, based on former research, effectively improve Wiener’s attack and verify its feasibility using the optimal asymptotic fraction of continued fraction.