In the context of functional encryption (FE), a weak security notion called selective security, which enforces the adversary to complete a challenge prior to seeing the system parameters, is used to argue in favor of the security of proposed cryptosystems. These results are often considered as an intermediate step to design adaptively secure cryptosystems. In fact, selectively secure FE schemes play a role of more than an intermediate step in many cases. If we restrict our attention to group-based constructions, it is not surprising to find several selectively secure FE schemes such that no successful adaptive adversary is found yet and/or it is also believed that no adaptive adversary will be found in practice even in the future. In this paper, we aim at clarifying these beliefs rigorously in the ideal model, called generic group model (GGM). First, we refine the definitions of the GGM and the security notions for FE scheme for clarification. Second, we formalize a group-based FE scheme with some conditions and then show that for any group-based FE scheme satisfying these conditions we can reduce from its selective security in the standard model to adaptive security in the GGM, in particular, regardless of the functionality of FE schemes. Our reduction is applicable to many existing selectively secure FE schemes with various functionalities, e.g., the FE scheme for quadratic functions of Baltico et al. (CRYPTO, 2017), the predicate encryption scheme of Katz et al. (J Cryptol in 26:191–224, 2013), and Boneh and Boyen’s identity-based encryption scheme (EUROCRYPT 2004).

在功能加密 (FE) 的上下文中，一种称为选择性安全的弱安全概念强制对手在看到系统参数之前完成挑战，用于支持所提议的密码系统的安全性。这些结果通常被认为是设计自适应安全的中间步骤密码系统。事实上，在许多情况下，选择性安全的 FE 方案所起的作用不仅仅是中间步骤。如果我们将注意力限制在基于组的结构上，那么找到几个选择性安全的 FE 方案就不足为奇了，这样还没有找到成功的自适应对手和/或也相信即使在实践中也不会发现自适应对手未来。在本文中，我们旨在在称为通用组模型 (GGM) 的理想模型中严格阐明这些信念. 首先，我们细化了 GGM 的定义和 FE 方案的安全概念以进行澄清。其次，我们将具有某些条件的基于组的 FE 方案形式化，然后证明对于满足这些条件的任何基于组的 FE 方案，我们可以从标准模型中的选择性安全性减少到 GGM 中的自适应安全性，特别是，无论FE方案的功能。我们的减少适用于许多现有的具有各种功能的选择性安全有限元方案，例如，Baltico 等人的二次函数的有限元方案。(CRYPTO, 2017)，Katz 等人的谓词加密方案。(J Cryptol in 26:191–224, 2013)，以及 Boneh 和 Boyen 的基于身份的加密方案 (EUROCRYPT 2004)。