自1979年以來，Shamir提出(k,n)秘密分享後，具門檻式的秘密分享的研究便開始受到廣泛地研究。(k,n)秘密分享是將秘密分成n個子秘密，從任k個或k個以上的子秘密還原出原始秘密。倘若子秘密的數目低於k，則無法獲得任何秘密資訊。把秘密分享擴展到秘密影像分享時，並不只是簡單地把秘密替換成數位影像。而是要考慮下面四個主要關鍵問題：(1)秘密影像分享的秘密需是有意義的影像內容(2)有限域的選擇(3)多項式哪裡嵌入秘密(4)子影像內容是隨機或有意義。本論文主要是研究第二項—有限域的選擇，以維持還原後的秘密影像品質。
以有限域的選擇而言，很明顯地秘密分享需要選擇模除運算的mod p有限域，而且這個質數p必須是大於秘密值的最小質數。另一方面，秘密影像分享需能處理0~255的像素值。因此，最合理的有限域選擇是GF(2^8)有限域，它可以處理影像中的256個灰階值可以使秘密影像無損地還原。但是使用GF(2^8)有限域，運算較為複雜、耗費的時間也隨之提升。有人採用mod 251這種簡單的模除運算。因為251是小於256的最大質數，所以mod 251是有限域。雖然選擇mod 251降低了運算複雜度，但是大於250的像素值需刪減至250以下，這會導致失真。
本論文，我們保留了簡單模除運算在計算上帶來的優勢，並且放鬆植基於像素的處理方式(例如：1個像素是N=8位元、而2個像素是N=16位元)。本篇論文使用小於2^N的最大質數PG以及大於2^N的最小質數PS做為模數。使用PG時，若數值不小於PG則刪減像素值。使用PS時，需配上更改與重算策略來調整生成的模數運算結果，使其得落在合法的範圍0到2^N－1之間。更改時，可能會產生失真。在 N=8~17時，我們理論分析出最佳使用的PG與PS值。

Since Shamir proposed the (k, n) secret sharing scheme in 1979, research of threshold secret sharing has been widely studied. (k, n) secret sharing divides the secret into n shares (or shadows), and reconstructed by using any k or more shadows. However, nothing is revealed about secret for less than k shadows. When extending secret sharing to secret image sharing (SIS), we have to consider the following issues: (i) the secret of SIS should be meaningful visual image (ii) the choice of finite field (iii) embedded positions in a polynomial (iv) meaningful or unmeaningful shadows. In this thesis, our study is dedicated on issue (ii) how to choose a finite field to maintain the visual quality of recovered image.
A finite field of secret sharing should be used, such that the reconstruction can be successfully accomplished. Obviously, for achieving simple and fat computation, we often choose mod p modular operation, where p is a prime and larger than secret value. For SIS, we have to deal with the pixel values from 0 to 255. The most reasonable choice is GF(2^8), which can process 256 grayness in image and recover secret image without distortion. However, this increases the computation complexity when using GF(2^8). Thus, using simple modular arithmetic operation, e.g., mod 251, is adopted. Note: 251 is the greatest prime less than 256. This modular arithmetic has the less complexity when compared with GF(2^8), but has distortion because the grayness 251~255 is truncated to 250.
In this thesis. We still use simple modular arithmetic for efficient computation, and meantime we loose the restriction of using pixel-wised operation (for example: 1 pixel is N=8 bits and 2 pixels is N=16 bits). Here, we use the greatest prime PG, less than 2^N or the smallest prime PS greater than 2^N. For using mod PG operation, if the pixel value is not lesser than PG then we should truncate the pixel value. For using mod PS operation, the modify-and-recalculate strategy will be used to make sure that the modular result is within the interval [0, 2^N－1]. The distortion could be generated from modification. For N=8~17, we theoretically and experimentally to analyze the optimal using of PG and PS for various values of N.

Chapter 1 Introduction 1

Chapter 2 Related Work 5
2.1 Polynomial based SIS 5
2.2 Wu et al.’s method 6

Chapter 3 The Proposed Method 7
3.1 Motivation 7
3.2 Framework 8
3.3 Modify-and-recalculate strategy 11
3.4 Image quality by mod PS sharing 15

Chapter 4 Experiment and Comparison 19
4.1 Number of flipped bits 20
4.2 Visual quality 23
4.3 Comparisons 25
4.4 Hybrid used of mod PS and PG 28
4.5 Security analysis 28
4.6 Computational complexity 30

Chapter 5 Conclusion and Future Work 33

References 35

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