(k, n) 秘密影像分享(Secret Image Sharing; SIS)是一種門檻機制的影像分享方式，將一個祕密影像分成n個子影像，並且再分送給每個參與者。最常使用的SIS機制是植基於多項式的秘密影像分享(Polynomial based SIS; PSIS)。解密時，若參與的子影像數量高過門檻值k，PSIS機制就能使用多項式插值計算解回秘密影像。多項式插值運算是基於有限域的計算，為了確保解回無失真的秘密影像最合理的有限域選擇是GF(28) (註:以處理一個像素256個灰階值為例)。有些研究則是採用簡單的模除計算代替GF(28)降低計算複雜度，但是簡單模除計算解回的影像會失真。本論文繼續研究簡單的模除運算以保留低複雜度的計算。為提高影像的解回品質，我們以多質數模除運算代替單一質數模除運算，建構一個多質數模除計算的SIS (SIS with Multi-Prime Modular Arithmetic; SISw/M)機制。

我們的SISw/M每次處理秘密影像的N個位元數不限是一個像素(N=8)。令PG是小於2N的最大質數、PS是大於2N的最小質數、NG – 2N-1是小於2N的最大整數、NS – 2N-1是大於2N的最小整數。我們的SISw/M使用介於(PG, NG ]的多質數所構成的複合數N’G，或介於[PS, Ns ]的複合數N’S。這樣選擇的N’G和N’S與2N差異，會比PG 和PS與2N的差異還小，所以解回的影像失真最小。複合數中的最小質數需儘量大，這樣所完成的(k, n)-SISw/M機制有大的“n”值才能有實際應用價值。

論文主要結果是對N=8~24位元找到最適合的 和 (註:其中複合數的最小質數是選擇7)，若是沒有符合的N’G和N’S則PG和PS單一質數就是較佳的選擇。另外，由於我們 (k, n)-SISw/M的多項式插值運算是基於模除一個複合數，所以可使用中國餘數定理來做漸進式解密。理論分析、和實驗結果證明了我們的(k, n)-SISw/M優點: 提高了解回影像的PSNR、並能做漸進式解密。

A (k, n) Secret Image Sharing (SIS) is a threshold scheme to share secret image, where an image is subdivided into n shadow images and delivered to involved participants. The most popular SIS approach is polynomial based SIS (PSIS). For recovery, if the number of involved participants is greater than k, the secret mage can be recovered via polynomial interpolation. The calculation of polynomial interpolation is based on finite field. Obviously, when dealing pixel-wise image the best choice is GF(28) finite field for lossless recovery. Some researches adopt simple modular arithmetic instead of GF(28) to reduce the computation complexity. In this thesis, we continue using simple modular arithmetic to retain the simple computation complexity. Meantime, to recover secret image with high visual quality we use multi-prime modular arithmetic instead of single prime to design a SIS with multi-prime modular arithmetic (SISw/M).

We deal with every N bits in an image. Let PG be the greatest prime less than 2N, PS be the smallest prime large than 2N, NG = 2N-1 be the greatest integer less than to 2N, and NS = 2N-1 be the smallest integer large than to 2N. The proposed SISw/M uses the multi-prime composite number N’G ∈[PG, NG ] or the multi-prime composite number N’S ∈[PS, Ns ]. The differences |N’G - 2N | and |N’S - 2N | are smaller than |PG - 2N | and |PS - 2N |, respectively. Thus, using N’G and N’S for polynomial interpolation have the less distortion of recovered image. Meantime, the small prime of N’G and N’S should be large such the value of n in (k, n)-SISw/M can be realized as large as possible.

Our main contribution is to figure out all the values of N’G and N’S for 8N24 (note: the small prime is 7 in these composite numbers). If we cannot find N’G and N’S, then sing prime PG and PS are best choices. In addition, the polynomial interpolation is based on modular composite number, we can use Chinse remainder theorem for progressive recovery. Theory analyses and experiments show the advantages of our SISw/M: the high PSNR of recovered image and the progressive recovery.

Chapter 1 Introduction 1
Chapter 2 Previous Works 5
2.1 The (k, n)-SIS 5
2.2 The (k, n)-SIS Using (mod PG) and (mod PS) 6
2.3 Chinese Remainder Theorem 6
Chapter 3 Motivation and Design Concept Using Multi-Prime Modular
Chapter 4 The Proposed (k, n)-SISw/M 15
4.1 Choosing N’G and N’S for Minimum-Loss Reconstruction 15
4.2 Using CRT for Progressive Reconstruction 17
4.3 Shadow Generation and Secret Reconstruction of The 24
(k, n)-SISw/M 24
Chapter 5 Experiment and Comparison 35
5.1 Recovered Image Quality 35
5.1.1 Comparison of using N’S and PG 36
5.1.2 Comparison of using N’S and PS 38
5. 2 The Progressive Recovery of (k, n)-SISw/M using CRT 43
Chapter 6 Conclusion and Future Work 47
References 48

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