密碼系統一般可分為區塊加密器(block ciphers)和串流加密器(stream ciphers)兩種。在此,使用渾沌系統之擬亂數產生器的渾沌式串流加密器(chaotic stream ciphers)已經被提出。一個渾沌式的動力系統意謂著在相空間中的數值軌跡演化會展現非週期性、複雜、和對初值敏感的現象。根據Lorenz系統的渾沌性質,我們詳述一個簡單的亂數產生器。利用適當的模運算處理, Lorenz系統產生之亂數可用來當作加解密金鑰(key)。在本篇論文中,我們將示範利用線性同餘方式產生亂數是不安全的,並且分析Logistic映射之安全性。透過統計測試的方式,且和Rong He及P.G. Vaidya 兩位學者所提出的密碼系統作比較,我們發現使用Lorenz系統的渾沌式串流加密器(CSC)有容易完成、高隱密性、高效率、安全性高可抵抗竊密者等優點。

Cryptographic systems are generally classified into block and stream ciphers.The cryptosystem CSC, chaotic stream ciphers, of combining a pseudo-random number generator of chaotic system with classic cryptography has been presented. A chaotic dynamical system means that the numerical trajectory in the phase space exhibits the phenomenon of aperiodic, complicated,
and sensible dependence on initial conditions. We describe a simple random number generator based on the chaotic property of the Lorenz system. The random numbers generated by the Lorenz system can be used as secret keys for encryption and decryption after suitable modulo operations have been applied. In this thesis, we will illustrate that the linear congruential recur-
rence is not secure and also analyze the secrecy of the logistic map. Through statistical tests, the CSC cryptosystem using chaotic Lorenz system as compared to that developed by Rong He and P.G. Vaidya has the advantages of easy implementation, good privacy, efficiency, and is robust against intruder.

Contents
1 Introduction 1
1.1 Cryptography . . . 1
1.2 Chaotic cryptosystem . . . 5
1.3 Organization of this work . . . 6
2 The pseudo-random number generator 7
2.1 Introduction . . . 7
2.2 The linear congruential pseudo-random number generators . 9
2.3 The statistical tests of random number generators . . . 9
2.4 The theoretical analysis of statistical random number
generator tests . . . 21
3 The Lorenz equation and its potential use for encryption 26
3.1 Introduction 26
3.2 Derivation of the Lorenz equation . . . 27
3.3 One point of view for extracting messages masked by chaos 38
3.4 The pseudo-random number generator using Lorenz equation 43
4 The CSC cryptosystem 71
4.1 Introduction . . . 71
4.2 The algorithm to encrypt and decrypt . . . 72
5 Conclusion 85

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