Shamir秘密分享是將秘密分享成n個子秘密，並且可以從任何k個子秘密恢復原始的秘密。但是，不能從（k-1）或更少的子秘密獲得任何秘密相關信息。將秘密分享使用在數位影像，就是所謂的秘密影像分享。它的子秘密與原始秘密都是數位影像。 (k, n) 秘密影像分享 (簡稱為(k, n)-SIS) 是結合了密碼學和影像處理的特性，讓影像能夠在分享的過程中被保護，但影像被保護的同時可能必須付出還原影像時品質嚴重降低的代價，因此，如何提升安全性，以及如何使加密後的影像在復原時能夠保留更高度的視覺品質，是重要的研究方向。
當秘密分享擴展到秘密影像分享如何選擇是適當的有限域，而不會造成影像的失真。植基於多項式的(k, n)-SIS一般使用GF(251) 有限域。這個簡單的(mod 251) 模除運算可以很快速完成。在GF(251)中我們只能夠計算0到250的數值，所以秘密影像的像素值必須被調整為介於 0 ~ 250之間的數值而造成失真。若要達到完原無損的回復秘密影像，就得使用GF(2^8) 有限域。但是基於多項式的有限域，比簡單的模除運算要花費較多的時間。
為了改善這個問題，本篇論文提出了以模除計算的(k, n)-SIS但是不再植基於像素而是每次處理N 個位元，以提升回復的秘密影像品質。因為是使用簡易模除，仍然保有計算效率。在本論文，我們理論分析了N 的最佳解。由實驗結果，說明了這些最佳解確實大幅提升了影像品質幾近無失真。

Shamir’s secret sharing divides the secret message into n shares. The secret can be reconstructed by using any k shares, but (k-1) or less than k shares have no information about the secret. When applying secret sharing on digital image is called as secret image sharing (SIS). A (k, n)-SIS has the threshold property of secret sharing and meantime deals with image processing to achieve image security. However, most (k, n)-SIS schemes may degrade the visual quality of recovered secret image. Therefore, how to achieve security and high image quality of secret image deserve studying.
In general, when secret sharing extended to secret image sharing, we have choose a finite field and cause the less distortion in recovered image. In polynomial based (k, n)-SIS, the GF(251) finite field is often used. The operation of GF(251) is a simple modular function, and can be easily finished. However, in GF(251), we can only deal with the pixel values from 0 to 250. Thus, the grayness 251~255 should be truncated to 250, and this result in image distortion. If we want to recover a distortion-less secret image, a polynomial based GF(2^8) finite field should be adopted. But, the operation of GF(2^8) is more complicated than the simple modular function.
A (k, n)-SIS based on simple modular arithmetic is proposed, which we use bit-wise operation instead of pixel-wise operation. We process N bits every time to enhance the recovered image quality. Because our method using simple modular arithmetic still has efficient computation. In this thesis, we theoretically give analysis to figure out the optimal values of 𝑁. Experimental results demonstrates using these optimal values of N may significantly enhance the visual quality of recovered image quality, and almost have no distortion.

Chapter 1 Introduction 1
1.1 Background 1
1.2 Contribution of the Thesis 3
1.3 Organization of the Thesis 4

Chapter 2 Preliminaries 5
2.1 Polynomial Based Secret Image Sharing Scheme 5
2.2 LSB Substitution and OPAP 6

Chapter 3 Proposed scheme 9
3.1 Motivation 9
3.2 Framework and Design 10
3.3 Implementation and Analysis 14
3.4 Choosing Optimal 𝑁 17

Chapter 4 Experiment and Discussions 19
4.1 Experimental Results 19

Chapter 5 Conclusion and Future Work 27

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