隨機性是密碼學中不可或缺的要素。在量子資訊發展以前，我們通常利用在缺乏足夠資訊下計算複雜的物理系統或解決困難的數學問題時的困難度，來產生難以預測的隨機數。根據量子理論的一個基本公理，測量一個量子系統得到的結果是機率性的。言下之意，隨機數可藉由測量量子系統產生。此外，基於量子理論中另一個特性—貝爾非局域性 (Bell nonlocality)，一個令人驚艷的概念—self-testing—以及與設備無關 (device-independent) 的框架被提出。藉由此框架，我們可以在對設備內部的機制一無所知的情況下，藉由設備產生的數據來推斷設備的運作情形。回到隨機數產生的話題，量子資訊讓我們得以在與設備無關的框架下來產生隨機數。考慮兩個裝置 (或量子系統)、兩種輸入 (或量子測量) 以及兩種輸出 (或測量結果) 的情況下，我們可使用 Clauser、Horne、Shimony 和 Holt 提出的貝爾不等式 (Bell inequality)，也就是 CHSH(貝爾) 不等式來判斷有多少的隨機數可以從裝置的輸出中萃取出來。要取得隨機數，觀測到 CHSH 不等式的違反是一個必要條件；然而，僅僅藉由 CHSH 不等式的違反程度所推斷出的隨機數的量，在某些情況下過於寬鬆。若考慮所有輸入和輸出的資訊，理論上，我們可以從相同的實驗數據中得到更多的隨機數。這不禁讓人開始思考是否能改良原本只考慮 CHSH 不等式的量子隨機數生成協定。從 [34]以幾何學的角度探討量子非局域性的研究取得的靈感，我們嘗試在基於 CHSH 不等式的量子隨機數生成協定中加入額外的計分方式與對應的終止標準。在貝爾實驗的場景下，我們可以定義三種隨機量，分別考慮從 (1) 單邊或 (2) 雙邊的輸出萃取隨機數，以及 (3) 當其中一裝置不被信任時，從單邊的輸出萃取隨機數。我們將我們提出的協定套用至這三種隨機量。利用近期發展的一個有用的理論—熵累積 (entropy accumulation) 理論，和 Brown、Fawzi 與 Fawzi [26] 建構的一個在與設備無關的框架下計算條件熵的方法，我們可以將分析量子隨機數生成協定的困難問題轉換成可透過數值計算解決的問題。在此之上，我們提供一個分析量子隨機數生成協定的方法，包刮對協定安全性、漸進無限數據下的的隨機數萃取量、有限數據下隨機數萃取量和最少需要的數據量的分析。我們比較基於 CHSH 不等式的量子隨機數生成協定和我們提出的版本，發現我們的版本在隨機數萃取量 (無論無限或有限數據) 和最少需要的數據量都有優勢。

Randomness is an essential ingredient in cryptography. With classical resources, (pseudo-) randomness is generated from either a lack of information on complicated physical systems or the intractability of mathematical problems. One of the fundamental axioms of quantum theory indicates that the indeterminacy of measurement outcomes is generic, which implies that inherent randomness can be generated with quantum resources. Moreover, another elemental feature of quantum theory, Bell nonlocality, brings out the idea of "self-testing" and the device-independent (DI) paradigm, where the devices' behavior can be verified by the statistics it produces with a minimal number of assumptions. In the bipartite, two-input, and two-output Bell scenario, a violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality is necessary to certify randomness. However, the amount of randomness certified by solely the CHSH Bell violation is generally not tight. Motivated by the study of quantum geometry [34], we incorporate the zero-probability constraints into the CHSH-based DI randomness expansion (DIRE) protocol. In this construction, we explore the family of DIRE protocols for three notions of DI randomness: local, global, and blind. By applying the recently developed technique of entropy accumulation, together with the numerical method devised in [26], the difficulties in the analysis of DIRE protocols can be overcome by numerical optimizations. Based on this, a comprehensive analysis of the protocols' security and performance is provided. A comparison between the standard CHSH-based and our proposed protocols manifests the advantages of our proposal in many aspects, including better performance in both asymptotic and finite regimes.

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