隨著愈來愈多商業活動及資訊傳輸在網路上進行，許多數位簽章及加密技術開發隨之發展，以幫助驗證傳輸訊息的來源及完整性；而基於基本的數位簽章技術，各種滿足隱私保護需求的簽章原型也相繼被開發出來，如環簽名和盲簽名，分別用來確保簽名者的匿名性以及維護簽署訊息的內容保密性。在這種多簽名方案中，由陳（ESORICS' 94）首先提出，並且隨後由 左等人(ISPEC'08)正式定義的N取1模糊簽章（Oblivious signature）方案，不僅提供了進一步的信任的基礎，還能同時保有像盲簽名一樣保護簽名請求者隱私的特性。在該方案中，接收者首先選擇一組訊息，其中一個是接收者打算獲得簽名的訊息；接收者與簽名者執行完互動式簽章演算法後，接收者將能得到他事先選定的訊息的簽章，而就簽章者的角度而言，他只知道他確實簽了訊息集合中的某一個訊息，卻不能得知接收者真正希望得到的是哪一個訊息的簽章。然而，目前現存的所有模糊簽章方案都是建立在以數論為基礎的難問題上，正如Shor於1994年所提出的量子計算演算法，這些基於數論難題所設計的方案將無法承受來自量子電腦的攻擊。為了克服這個問題，本篇論文提出了一個簡單的基於晶格（Lattices）的抗量子攻擊N取1模糊簽章方案，並且提出了安全性證明，證明其在隨機預言模型（Random oracle）下，基於晶格的難問題－最小整數解問題（Shortest integer solution problem），滿足強不可偽造性及模糊性。據我們所知，我們是第一個提出基於晶格的抗量子攻擊的模糊簽章方案。

As business activities and information exchange increasingly move online, digital signatures, among other cryptographic techniques, have been developed to help authenticate the source and integrity of digital information when transferred. Various types of signature primitives, such as ring signatures and blind signatures, have been introduced to satisfy privacy protection needs spanning from ensuring anonymity of a signer to maintaining secrecy of the content to be signed from a signer. Among different signature schemes, the 1-out-of-𝑁 oblivious signature scheme, which was introduced by Chen (ESORICS’ 94) and later formalized by Tso et al. (ISPEC’ 08), provides a further basis of trust while preserving the signature requestor’s privacy as blind signatures do. In this scheme, a recipient first selects a set of messages, one of which being the message he or she intends to obtain a signature for. After interacting with a signer, while the recipient will be able to obtain a signature on the predetermined message, the signer only knows that he or she signed one of the messages but remains oblivious to exactly which message was signed. However, all existing oblivious signature schemes are built upon the hardness of number-theoretic problems, which, as Shor demonstrated in 1994, cannot withstand attacks from quantum adversaries. To address this problem, this work proposes a novel quantum-resistant 1-out-of-𝑁 oblivious signature scheme based on SIS hard assumption. We also provide security proofs to demonstrate that the security requirements of ambiguity and strong unforgeability are satisfied under the random oracle model. To the best of our knowledge, the proposed scheme is the first 1-out-of-𝑁 oblivious signature that is secure against quantum adversaries.

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