(k, n) 秘密影像分享(Secret Image Sharing; SIS) 是一種加密技術，這種加密技術可以將一張秘密影像加密成 n 張子影像 (Shadow)並分給 n 位參與者。在秘密重建階段時，必須至少 k 位參與者提交子影像才能還原秘密影像。如果在秘密重建階段時，參與解密的 k 位參與者中有一位不誠實的參與者提交偽造的子影像，而這會造成秘密無法在秘密重建階段被還原出來。然後，不誠實的參與者可使用其他參與者的子影像以得到秘密影像。由於無法解回秘密，這時候秘密影像會被註銷Dealer 會再重新分享新的秘密影像給 n 個參與者。雖然，此時不誠實的參與者僅能得到被註銷的秘密似乎無任何資安疑慮，但是這仍然會有安全風險。原因說明如下: 假如 Dealer 欲分享一個重要且緊急的秘密影像 (例如: 軍隊的作戰地圖)給其他 n 位參與者，而此時由於不誠實參與者的關係導致秘密影像被註銷，因為緊急所以Dealer 仍然要分享原來的秘密訊息，所以他會將原始的秘密影像修改過後重新分享。總而言之，新的秘密影像與舊的註銷秘密影像可能有相關性。即使新的秘密影像和被註銷的秘密影像毫無關聯，但是本來欲分享的秘密資訊洩漏本身就是嚴重的資安風險，我們稱這種提交偽造子影像以達到的攻擊為不誠實參與者攻擊(Dishonest Participant Attack; DPA)。從上面的敘述，研究可抵禦DPA 的秘密影像分享記制是有其必要性的。
本論文利用錯誤控制碼的錯誤偵測、及改正能力來實作可抵禦DPA 的植基於改錯碼的秘密影像分享(Code-based SIS; CSIS)機制。而我們所提出的方法不僅能克服DPA 攻擊，同時也可以解決SIS 的殘影問題。在秘密分享段時，我們會將秘密像素使用錯誤控制碼、和交織技術編碼成二位元BCH 碼，然後將其轉換成子影像。此時，Dealer 會利用一把密鑰對每個參與者的子影像像素進行循環移位。藉由計算癥狀、及癥狀解碼以偵測，我們可以識別偽造的子影像。若是參與者所提交的子影像都是正確的Dealer 才會公布密鑰，然後k 個參與者藉由密鑰還原子影像後以解回秘密影像。理論證明、及實驗結果皆顯示了我們 (k, n)-CSIS機制的有效性。

A (k, n) secret image sharing (SIS) is a cryptographic technology that encrypts a secret image into n shadow images (referred to as shadows) and delivered to n participants. For recovery, any k participants can provide their shadows to reconstruct secret image. Suppose that a dishonest participant among the k involved participants submits a fake shadow, and the secret image cannot be recovered. For the case, the
dishonest participant may privately recover secret by his own correct shadow with other shadows. However, when the secret cannot be recovered, the recovery process is stopped and the secret image is revoked. Then, the dealer will reboot a new sharing phase with a new secret image. It seems that the dishonest participant can only obtain
a revoked secret image, and this has no security concern. However, this still has a security risk. The reason is described as follows. Suppose a dealer wants to share a secret image containing a critical and urgent information (such as a battle map for the army). And, the secret image is revoked due to a dishonest participant is involved in reconstruction. Because the secret information is urgent and has to be shared to other participants, the dealer may modify the revoked secret image and reboot sharing phase with a new secret. Therefore, the new secret may be related to a revoked one. Even though the new secret image is not related to the revoked secret image, the leakage of
secret information is a critically security risk. We herein call this attack providing fake shadow as dishonest participant attack (DPA). The above description implies that there is a need to propose the (k, n)-SIS to address DPA.
In this thesis, we propose a code-based SIS (CSIS) based on error correction code (ECC) with error detection and correction capabilities. The proposed (k, n)-CSIS not only overcomes DPA, but also tackles the residual image problem (RIP) in polynomial based SIS (PSIS). In secret sharing phase, we first encode the secret pixels by ECC and interleaving technology into BCH codes and then convert them into shadow pixels. After that, the dealer will use a key to cyclically rotate shadow pixels. In verification phase and identification phase, we use syndrome computation to verify and identify faked shadows and dishonest participants. If all involved shadows are correct, the dealer
publicly announces the key, on which participants may obtain the original shadows for recovering secret. Theoretical derivations and experimental results demonstrate the effectiveness of the proposed (k, n)-CSIS.

Chapter 1 Introduction....................................................... 1
1.1 Background .....................................................1
1.2 Organization of The Thesis .....................................2
Chapter 2 Observations on Constructing CSIS ........................5
2.1 CSIS based on Matrix Theory ....................................5
2.2 CSIS Using Reed Solomon (RS) Codes .............................6
Chapter 3 The Proposed (k, n)-CSIS .................................9
3.1 Critical Problems ..............................................9
3.2 The Proposed (k, n)-CSIS.......................................12
3.3 Theoretical Analysis ..........................................20
3.4 Numerical Examples ............................................22
Chapter 4 Experiment and Comparison................................35
4.1 Experimental Results...........................................35
4.2 Discussion and Comparison .....................................40
Chapter 5 Conclusions and Future Work .............................47
References ........................................................49

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