混沌系統具有不規則，非周期性，無法預測和對於初始條件相當敏感的特質。而這些特質符合與密碼學上混亂和擴散的特性。因此，近年來混沌系統在密碼學上的應用被廣泛的討論與研究。然而，當混沌系統數位化的過程中，原先所保有的混沌特質產生了變化，這種現象又稱為動態特性的降低。一個明顯的例子就是數位化的混沌系統容易產生出一個短周期的輸出軌道。而一個短周期的軌道在統計學的角度上則是容易被分析且不適合應用於密碼系統。在這一篇論文中，我們將研究動態退化的現象並且提出數個方法來提升數位化混沌系統的隨機品質。主要的研究內容如下。首先，我們提出了強化型邏輯映射混沌系統。此系統擁有比傳統邏輯映射混沌系統更大範圍的可用參數，而且這個可用參數範圍內不會存在短周期的參數。基於強化型邏輯映射混沌系統，我們更提出了強化型多維度混沌系統，使其具有更多的可用參數來應用於安全傳輸系統。第二，我們提出了變化型邏輯映射混沌系統。此系統明顯的增加了單位時間的輸出量與隨機品質。此外我們串接數個變化型邏輯映射混沌系統來建立多變化型邏輯映射混沌系統，使其可容易擴張，並可以快速的產生具有高複雜度與長週期特性的混沌數列。最後，在本論文的第三部分則是針對偽隨機變數產生器應用提出了數位化變更型邏輯映射混沌系統。在這個系統中我們使用了參數選擇與擾動技術，我們有效的減少了系統的計算量並提高了輸出的複雜度。在現行的偽隨機亂數數列測試平台測試結果顯示，相比於先前所提出的混沌偽隨機亂數產生器，我們的系統使用了較低的硬體成本產生了較高隨機品質的偽隨機數列。

The orbit of a chaotic system is irregular, aperiodic, unpredictable, and sensitive to initial
conditions. These characteristics coincide with the confusion and diffusion properties in
cryptography. In recent years, chaotic systems have been studied for secure communications.
However, when a chaotic system is digitalized, it results in some unexpected behaviors
due to limited precision. It is known as dynamical degradation. An obvious phenomenon
is that an orbit enters a cycle with unpredictable length. The orbit with short cycle length
has poor quality of randomness because it can be easily analyzed from statistical point of
view.
In this dissertation, we focus on improving quality of randomness for digitalized logistic
maps. New modified logistic map and techniques are proposed to improve the degree of
complexity for secure communications and pseudo random number generation. The main
achievements of this dissertation are as follows.
First, we propose a Robust Logistic Map (RLM) which has a larger parameter space
than classical logistic map. Moreover, there are no windows with short period-length in the
parameter space. Based on RLM, a Robust Hyper-Chaotic System (RHCS) is constructed
for secure-communication systems with large parameter space.
Second, we propose a Variational Logistic Map (VLM) to significantly increase the
throughout and quality of randomness of RLM. Moreover, a Multiple Variational Logistic
Map (MVLM) is proposed for fast chaotic sequence generator. Because of the regular
architecture of MVLM, it is easy to scale up the system degree to provide long output
sequence with high degree of complexity and large key space for secure communications.
Pseudo Random Number Generators (PRNGs) are often an important component in secure
communications. In the third part of this dissertation, we propose a PRNG based on
a Digitalized Modified Logistic Map (DMLM). Two techniques, constant parameter selection and output scrambling are employed to reduce the computation cost and to increase the
complexity of the PRNG. Compared to previous digitalized chaotic systems based PRNGs,
our DMLM-PRNG has better quality of randomness and lower hardware cost.
Each of our system mentioned above has been implemented. Comparisons between
our systems and previous work are conducted in terms of hardware cost and throughput.
Moreover, the quality of randomness is demonstrated by statistical analysis.

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