秘密分享(SS)是一種方法，將秘密分享成多個子秘密給一群參與者以達到保護秘密資料。實現SS的作法通常是使用門檻式的(k, n)- SS其中k n，在n份子秘密資料中只要任k份就可以重建秘密資料，(k−1)份或更少子秘密則無法得任何秘密訊息。最常使用的SS方法是基於多項式的Shamir的 (k, n)-SS，這種方法最初的目的是為了保護密碼機制的金鑰。除了保護金鑰以外，(k, n)-SS還可以擴展到其他的領域，應用最廣的是秘密影像分享(Secret Image Sharing; SIS)。將SS擴展到SIS並不只是簡單地把秘密替換成影像，而是要考慮秘密的形式、有限域的選擇、嵌入秘密的位置、以及子影像的形式。例如，SS的秘密可以是隨機的因為它原本就是要保護金鑰，而金鑰當可以是雜亂隨機值。但是SIS的秘密是數位影像。一個視覺影像應是有意義的內容，所以它不能是雜亂的隨機影像。由於秘密形態的差異，所以不是所有SS方案都能全盤移植到SIS。
本論文的研究是秘密文件分享(Secret Document Sharing; SDS)。與SIS一樣，SS方案並不能直接擴展到SDS，而是要適當的修訂與設計。SDS首先由Tu和Hsu 提出，是將一份秘密文件藉由一份有意義的公開文件分享成n份雜亂的子文件。在Tu和Hsu的(k, n)-SDS中，任k份子文件、和公開文件能解回秘密文件。最近，Liu等人使用有限域GF(65537)代替Tu 和 Hsu的GF(257)，一次處理文件的2個內碼以加快文件分享與解密速度。本論文所提方案延續了Liu等人機制，但是設計了有意義子文件的(k, n)-SDS。我們考慮的應用情境是參與者參與了多個(k, n)-SDS機制，此時他會持有來自不同(k, n)-SDS的子影像。若子影像是雜亂難以區分，會造成管理的不變。我們有意義子影像的(k, n) -SDS可以解決子影像管理問題，另外這些有意義的子影像也可用於傳送額外的訊息給參與者。除此之外，我們機制藉由全有或全無轉換(all-or-nothing transform)使用了多項式中的所有係數來嵌入內碼並可達到安全性。
我們的方案與Tu和Hsu的SDS、Liu等人的SDS另外一個不同的地方是沒有一個有意義的公開文件。若考慮公共服務公告(Public Service Announcement; PSA)的應用情境，要向所有參與人員公開一些信息。這個 PSA 應用場景就需要一個有意義公開文件的(k, n)-SDS，本文也提出將我們的 (k+1, n+1)-SDS修改為一個有意義公開文件的(k, n)-SDS以做PSA應用。最後，我們將現有的SDS方法與所提的方案進行了比較呈現了我們的優點。

Secret sharing (SS) could protect and share secret data into multiple sub secrets among a group of participants. SS is usually realized as a threshold (k, n)-SS, where k( n) is the threshold, any k shares can reconstruct the secret but (k−1) or fewer shares learn nothing about secret. The most common method to realize SS is the polynomial based Shamir’s (k, n)-SS, which is to safeguard the secret key in a cryptosystem. Except protecting secret key, SS can also be extended to various research fields. The most popular research is secret image sharing (SIS). Extending SS into SIS is not trivially using digital image as secret. Some issues need to be carefully considered: (i) the secret type, (ii) the choice of finite field, (iii), where to embed secret in a polynomial, and (iii) the pattern of shadow image. For example, the secret type in SS could be random because the secret is a “key” in cryptosystem. However, the secret of SIS is a digital image. Obviously, a visual image should have meaningful content, and thus it cannot be a random image. The above implies that SS cannot be extended to SIS directly.
The thesis is about secret document sharing (SDS). Same as SIS, SS cannot be directly extended to SDS. The approach of SDS should be carefully designed. SDS was first proposed by Tu and Hsu, which a secret document is shared into n noise-like shares with a meaningful public document. In Tu and Hsu’s (k, n)-SDS, any k shares with the public document are collaboratively used for recovering secret document. Recently, Liu et al. use GF(65537) instead of Tu and Hsu’s GF(257) on two inner codes each time to speed up sharing and recovering. In the thesis, we continue to use Liu et al.’s method, but design a (k, n)-SDS with meaningful shares. Our SDS has the easy management of shares applied on the scenario that a participant involved in various (k, n)-SDSs, and the participant has to hold several shares. If these shares are noise-like, they are indistinguishable and not easily managed. Our (k, n)-SDS with meaningful shares resolves this management problem. Meantime, the meaningful shares can also be used to deliver additional information to respective participants. In addition, all-or-nothing transform is used in our SDS, such that we can use all coefficients in a polynomial and meantime achieve security.
Another difference between our SDS and other two SDSs (Tu and Hsu's SDS, and Liu et al.’s SDS) is that we do not have a meaningful public document. Considering the Public Service Announcement (PSA) application, some information need to be public to all participants. In this PSA application, we need a (k, n)-SDS with meaningful public document, in the thesis, we also modify our (k+1, n+1)-SDS as a (k, n)-SDS with meaningful public document. Finally, we compare existing SDSs with the proposed SDS to demonstrate our advantages.

Chapter 1 Introduction　　 1
Chapter 2 Previous Works　　 5
2.1 Shamir’s SS　　 7
2.2 Previous SDS　　 8
Chapter 3 Motivation　　 13
Chapter 4 The Proposed (k, n)-SDS　　 19
4.1 Design Concept　　 19
4.2 Sharing and Reconstruction　　 31
4.3 Using Proposed Scheme as (k, n)-SDS with Meaningful Public Document　　 34
Chapter 5 Experiment and Comparison　　 37
5.1 Experimental Results　　 37
5.2 Comparison　　 45
Chapter 6 Conclusion and Future Work　　 51
References　　 53

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