秘密影像分享領域，Thien和Lin等人使用Shamir秘密分享的方法來分享祕密影像，將像素值嵌入 (k-1) 次方多項式的係數來分享秘密影像。大部分的SIS機制是討論一個群組內參與者分享祕密，不同於現有的SIS機制，本論文"植基於同態特性的多項式組間和內秘密影像分享機制"，不只討論傳統SIS的組內分享密影像，也討論群組之間的組間分享密影像。本篇論文提出一種用於多組群體的組間 (Intergroup) 及組內 (Intragroup)的 (k, n, m) 秘密影像分享機制 (Secret Image Sharing; SIS) (簡寫成 (k, n, m)-I2SIS)，此處k值為解密的門檻值、m為群組數、n為參與者的數目，這個機制可以讓(2m-1)張秘密影像在組間以及組內進行分享。
我們以一個例子來說明我們I2SIS機制的運作與應用情境。所謂的群體通訊是由組內通訊 (同一群體內的通訊) 和組間通訊 (不同群體內之間的通訊) 所組成。I2SIS就是一種群體通訊機制，包含了組內和組間通訊。我們以一個使用 I2SIS軍事應用中的聯合作戰場景，包括陸、海和空軍事行動的三棲作戰為例來說明。此I2SIS例子m是3 (陸、海、空三軍)。這些軍事作戰地圖包含了陸軍(G1)、海軍(G2)、和空軍(G3)的各自軍種作戰圖。現代戰爭常需要聯合作戰。因此，我們也需要陸海(G1+2)、陸空(G1+3)、海空(G2+3)、甚至是(G1+2+3)三軍聯合作戰圖。所以，這個m=3的I2SIS機制需能分享3個組內秘密影像(G1、G2、G3)，及4個組間秘密影像 (G1+2、G1+3、G2+3、G1+2+3)。
我們的(k, n, m)- I2SIS機制結合了多項式SIS和同態性質 (註: 拉格朗日插值法有加法同態特性)，完成組合總數為 組間、組內的秘密影像分享。本論文的重點主要是影像的預處理，決定多項式數那些係數用於那一個組合來嵌入信息。我們提出兩種影像預處理的方法，並對秘密影像回復的比率進行理論估算。另外，多群組SIS (Multi Group SIS; MGSIS)與多層級SIS (Multi Level SIS; MLSIS)之間存在緊密的關係和高度的相似性。 從本質上講，這兩種方法是相同的。 他們只是使用不同的方式分享秘密影像。因此，我們的 I2SIS 的另一個應用場景可用於秘密影像的多層級保護。所以，與現有的SIS機制相比，我們的I2SIS機制有更多的應用價值。

Thien and Lin extend Shamir’s secret sharing to design secret image sharing (SIS) by embedding secret pixels into all coefficients of a (k-1)-degree polynomial. Most SIS schemes were dedicated to share one secret image within one group. Different to existing SIS methods, in the thesis "Homomorphic Polynomial Based Intragroup and Intergroup Secret Image Sharing", we discuss sharing secret image not only the intragroups but also in intergroups. A (k, n, m)-intragroup and intergroup secret image sharing (I2SIS) for multi-group is prosed, where k is the threshold, m is the number of group, and n is the number of involved participants. This (k, n, m)-I2SIS can be used for sharing (2m-1) secret images within groups (intragroups) and between groups (intergroups).
We briefly describe the notion of I2SIS busing an example. Generally, group communications are composed of intragroup communication (in the same group) and intergroup communication (among different groups). Our I2SIS is a kind of group communication. Here, we give a scenario of using I2SIS on military application. Consider joint operational warfare. Suppose a triphibious operation including military operations occurring on land, at sea and in the air, needs three groups of soldiers coming from army (G1), navy (G2) and air force (G3) involved in operation. Obviously, the qualified sets in G1, G2 and G3 can recover military operation maps for respective military branches: army, navy and air force. In modern warfare, there is a need of joint operation. Therefore, the qualified sets in G1+2, G1+3, G2+3, and G1+2+3 may recover army-navy, army-air force, navy-air force, and triphibious warfare combined operation map, respectively. Thus, this I2SIS wit m=3 can share 3 intragroup secret images (G1,G2,G3) and 4 intergroup secret images (G1+2,G1+3,G2+3,G1+2+3).
Our (k, n, m)-I2SIS combines polynomial-based SIS and homomorphism (note: Lagrange interpolation has additive homomorphism) to share (2m-1) secret images within intragroups) and between intergroups. Our maim issue is the preprocessing procedure that arrange and distribute coefficients of the polynomials for embedding secret pixels for some specific combinations of groups. We proposed two preprocessing approaches, and also theoretically evaluate the percentages of recovering secret image for each approach. In addition, there are tight relations and high similarities between multi group SIS (MGSIS) and multi level SIS (MLSIS). These two approaches are the same, and just share secret data using different ways. Thus, another application scenario of our I2SIS could be used in MLSIS. Thus, when compared with existing SIS schemes, our I2SIS definitely has wide applications.

Chapter 1 Introduction 1
Chapter 2 Previous Works 5
2.1. Polynomial based SIS 4
2.2. Additive homomorphism 5
Chapter 3 Motivation and Application 9
3.1 Motivation 7
3.2 Application Scenario of Using I2SIS: Military Application 11
Chapter 4 The Proposed (k, n, m)-I2SIS 15
4.1 Preprocessing Procedure 13
4.2 Sharing Procedure 18
4.3 Recovering Procedure 19
Chapter 5 Analysis and Extension of Preprocessing 25
Chapter 6 Experiment and Comparison 33
6.1 Experimental Results 30
6.2. Percentages of recovering secret images 39
6.3. Comparison 41
Chapter 7 Conclusion and Future Work 47
References 49

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