隨機動態加解密法

Randomly Dynamic Cryptosystem

劉伯彥(Liu Poyen)

指導教授 ： 黃淳權

亞洲大學/資訊學院/資訊科學與應用學系碩士班/碩士(2006年)

摘要

本篇論文中，我們所提出的隨機動態加解密法，在加密的過程中可分為編碼及加密兩個部份，明文於加密時，會先進行編碼再行加密，其中編碼係利用加密方所使用的固定密碼字典根據隨機時間函數產生的浮動密碼字典進行編碼，明文經編碼產生密碼文，再次根據時間函數進行加密，編碼及加密可視安全等級之需要，可單獨使用或是合併使用。
在加密系統中有一原典，原典將為不同的公司端產生相異的公司端密碼字典，公司端密碼字典將分配給其下的各個使用者一本獨一無二的使用者密碼字典即固定密碼字典，雖然加密時，使用者密碼字典會根據隨機時間函數再次產生僅供該次加密使用的浮動密碼字典，但於密碼文之解碼時，不會因為每個使用者其當次加密的密碼字典有所不同而無法進行反解密碼文得明文，因為在原典產生公司端密碼字典後，公司端密碼字典再為不同的使用者產生其獨特的使用者密碼字典，然而使用者密碼字典再根據隨機時間函數產生浮動密碼字典以供加密使用之以上所提的密碼字典間的轉換，皆是利用齊性轉換來完成的，因此即使是加密方及解密方的密碼字典不同，但彼此之間仍可進行明文及密碼的加解碼。
當使用者要將明文加密時，將由使用者密碼字典根據隨時函數產生浮動字典，對明文進行編碼，使明文轉變為密碼文，進而隨機時間函數也決定一加密協定，並利用浮動密碼字典取得加密引數(幅角)，對密碼文進行加密轉換，使密碼文變為密文，此時加密方將加密金鑰傳送至公司端，加密金鑰包含編碼及加密的所使用的引數和協定，此時公司端會根據解密方的使用者固定密碼字典產生解密金鑰，此解密金鑰僅對合法的解密方產生作用，因為解密金鑰需搭配解密方的使用者固定密碼字典及該份密文才能產生效用，因此解密方取得解密金鑰和密文時，才能根據解密金鑰和其所持有的使用者固定密碼字典對密文進行解密，但為了防止加密方透過長期傳送特定內容及長期蒐集及分析公司端傳送給解密方的加密金鑰，進而去猜測解密方的固定密碼字典的結構，雖然破解者最多僅能靠此方法猜測出密碼字典的部份內容，當破解者取得密碼字典的部份內容，破解者便可能破解其它使用者傳送至被破解者的部份密文內容，因此為了更高的保密等級的需要，因此在此也可透過隨機時間函數的方式，以產生供解密用的浮動字典用以隱藏使用者固定密碼字典。
密碼字典的結構或為一立體洋蔥球的設計，而齊性轉換具有保長、保角即保形的特點，在密碼字典為一洋蔥球的立體結構下，任何特定兩點，在經密碼字典轉換後，兩點間的絕對距離是不變的，進而任何特定三點間的角度、形狀也具有保持固定的特點，因此在加解密的演算上有其便利性及精簡性並得以互相進行轉換，齊性轉換的方法眾多，在本論文中，我們將僅以Euler轉換為我們的實施例加以說明。
此外，在本文中也將介紹我們所提出的加密三原則：1. Without Knowing what to find.(不知道尋找何物)2. Without knowing where to find or the space is too huge to search.(不知去何處找)3. How many to find? (找幾個？)，並探討破解者及駭客攻擊本法的各種可能情形，本文並討論在我們所提出加密三原則下，破解者及駭客為何無法進行有效率的攻擊，並針對我們所提出的加碼暨加密方法中，唯一可能遭到較大威脅的攻擊手段，即當加密者為破解者或駭客，並試圖破解解密方的密碼字典時，我們所設立的防範機制及其被破解的可能性的探討。
此外，文中也將指出，我們所提的隨機動態加解密法在對明文進行編碼時擁有 種加碼變化，但對密碼文加密時，則具有無限多種的方法來進行加密，並對於加解密過程中加密運算及解密運算的精度加以探討，雖然在加密運算及解密運算上若無更進一步的方法，若幅角被分割1024個等間隔時，在運算上可能需要小數點後20位數的精度，但對於現今計算機的運算能力並不構成太大問題，尤其所涉及的運算過程中僅是加法、減法及乘法的計算；雖然加密的過程中是利用矩陣進行的，但當密文解密時是不需要求其逆矩陣的，因為解密所需的運算，僅是一個轉置矩陣便能達成反解密文的運算，所以解密運算的步驟極短，相對地計算上的誤差累積極小。若採用離散性之計算，對於每一三元有序對之分量，第一分量為 ，第二分量為 ，第三分量則為 的整數，在運算上僅是加法、減法及乘法之運算，也因此我們在此提出與DES不同方式之重排，以DES來說對此數量的重排將會是 的運算，然而我們的重排將會是 的重排運算，對於DES，計算機在計算 時其所佔用的資源將是很龐大的，然而對於我們的 的運算及其逆運算，計算機將只佔用很小的資源，而且對於安全性而言，不會因此減低，此外在運算過程中，皆是整數的運算，整數運算將不會產生誤差，且逆運算矩陣為構成該矩陣之成分矩陣本身次序相反所構成之合矩陣，因此計算機運算量將大幅降低，所以在我們的計算過程中，加密及解密很快速且僅佔用很小的系統資源的。

關鍵字

密碼字典 ； 浮動密碼字典 ； 隨機時間函數 ； 齊性轉換 ； Euler轉換

並列摘要

n this thesis, a randomly dynamic cryptosystem is proposed. The processes are composed of two major parts. The first is a process of encoding and the other is encrypting. The process of encoding can be applied alone, without encrypting process.
Originally, there is a codebook, called original codebook. The original codebook is transformed with homogeneous transformations into some business codebooks, each one is different from the others. Each business has one such unique business codebook. A business has many users. Each user has one unique codebook which is again transformed from the business codebook with a homogeneous transformation, which is called fixed codebook. Once a custom wants to make a plaintext into a ciphertext, that fixed codebook is again transformed into a floating codebook with a random time function. So every time, the floating codebook is different.
Accordingly, the floating codebook is used to transform the plaintext from ordinary language words or sentences into codes. Because such codes are not easy to solve, it is already to be transmitted and accepted as the ciphertext.
The codes are further encrypted with a protocol which randomly creates a function for transforming the codes into the ciphertext. Such function may also be the homogeneous transformation with some augments which are provided by the fixed codebook with a protocol. Clearly, the fixed codebook is again used to release its subset with a protocol. Such a subset contains a series of codes and to be used as the augments. In this way the codes are transformed into more complex ciphertext.
At the receiver’s end, a key is received. Combine this key and the ciphertext, nothing but some codes are obtained. Again, the receiver has his own fixed codebook. With this fixed codebook, the above codes can be transformed to be the plaintext.
This article supposes the hackers can steal everything via the network. When the ciphertext and the key are intercepted by the hacker, the hacker can not solve it because the hacker does not have the fixed codebook of the receiver’s, which is never appeared in the network. In this article, a condition of a ciphertext sender may be the hacker to peep the fixed codebook of the receiver’s is discussed.
This article proposes the three principles of encryption, namely: 1, without knowing what to find, 2, without knowing where to find or the space is too huge to search, and 3, how many to find.
The encoding process has a huge sample space as large as . The encrypting process provides infinite possibility for adopting homogeneous transformations and possibilities in the augments. Therefore the principle 2 is satisfied.
In the method proposed, the ciphertext is nothing but 1 and 0’s. Without the protocol, an attacker can do nothing. No matter how hard he tries, what he can have is nothing but codes. Without the codebook, an attacker does not know even he get the correct codes. Hence the principle1 is satisfied.
We may adopt some mathematical method such as simultaneous equations to make an attacker can not but have to solve all the codes in a same time. This is also called the effect of avalanche. Firstly, the attacker should know the number of equations in a set of simultaneous equations. Suppose there are 100 sets of codes in the codes, the possibilities of partition is . The hacker doesn’t know the 100 and , the principle 3 is satisfied.
In performing the homogeneous transformations, Euler’s rotations are suggested. Further, this article suggests the possibility of onion structure of the codebook. In this way it is possible to have integer computation, actually permutation. Each set code in the codes is a triple ordered pair. Each component of the triple ordered pair shall be permuted during transformation. Or to keep the performance of the homogeneous transformations, Euler’s rotations are still hold. Each shell of the onion rotates independently with a pole of rotation.

並列關鍵字

codebook ； floating codebook ； homogeneous transformations ； random time function ； Euler’s rotations

參考文獻

[3] Diffie and M. E. Hellman, “New Directions in Cryptography,” IEEE Trans. Information Theory, vol.IT-22, no.6, pp. 444-654, Nov. 1976.

[5] Desai, “ The Security of All-or-Nothing Encryption: Protecting against Exhaustive Key Search,” Crypto 2000, Lecture Notes in Computer Science, vol. 1880, pp. 359-375, M. Bellare ed., 2000.

[6] M. Abdalla, Y. Shavitt and A. Wool, “Key Management for Restricted Multicast Using Broadcast Encryption,” IEEE/ACM Trans. Networking, vol. 8, no. 4, pp.443-454, Aug. 2000.

[7] H. Handschuh and S. Vaudenay, “A Universal Encryption Standard,” SAC’99, Lecture Notes in Computer Science, H. Heys and C. Adams eds., Springer-Verlag, vol. 1758, pp. 1-12, 2000.

[8] R. Canetti, S. Halevi and J. Katz, “Adaptively-Secure, Non-interactive Public-Key Encryption,” TCC2005, Lecture Notes in Computer Science, J. Kilian ed., Springer-Verlag, vol. 3378, pp. 150-168, 2005.