Let $S$ be the pool of $s$ parties and Alice be the dealer. In this paper, we propose a scheme that allows the dealer to encrypt messages in such a way that only one authorized coalition of parties (which the dealer chooses depending on the message) can decrypt. At the setup stage, each of the parties involved in the process receives an individual key from the dealer. To decrypt information, an authorized coalition of parties must work together to use their keys. Based on this scheme, we propose a threshold encryption scheme. For a given message $f$ the dealer can choose any threshold $m = m(f).$ More precisely, any set of parties of size at least $m$ can evaluate $f$; any set of size less than $m$ cannot do this. Similarly, the distribution of keys among the included parties can be done in such a way that authorized coalitions of parties will be given the opportunity to put a collective digital signature on any documents. This primitive can be generalized to the dynamic setting, where any user can dynamically join the pool $S$. In this case the new user receives a key from the dealer. Also any user can leave pool $S$. In both cases, already distributed keys of other users do not change. The main feature of the proposed schemes is that for a given $s$ the keys are distributed once and can be used multiple times. The proposed scheme based on the idea of hidden multipliers in encryption. As a platform, one can use both multiplicative groups of finite fields and groups of invertible elements of commutative rings, in particular, multiplicative groups of residue rings. We propose two versions of this scheme.

翻译：设$S$为$s$个参与方的集合，Alice为分发者。本文提出一种方案，允许分发者以特定方式加密消息，使得只有分发者根据消息选择的一个授权参与方联盟能够解密。在初始化阶段，流程中的每个参与方都会从分发者处获得一个独立密钥。要解密信息，授权联盟必须协同使用各自的密钥。基于此方案，我们进一步提出了一种门限加密方案。对于给定消息$f$，分发者可选择任意门限值$m = m(f)$。精确而言，任何规模至少为$m$的参与方集合都能计算$f$，而规模小于$m$的集合则无法完成此操作。类似地，密钥在参与方之间的分配方式可设计为：授权参与方联盟能够对任意文档进行集体数字签名。该基本方案可推广至动态场景，其中任意用户均可动态加入集合$S$。此时新用户将从分发者处获得密钥，任意用户也可离开集合$S$。在两种情况下，其他用户已分配的密钥保持不变。所提方案的核心特征在于：对于给定的$s$，密钥仅需分配一次即可重复使用。本方案基于加密中的隐藏乘数思想，其数学平台既可采用有限域的乘法群，也可采用交换环可逆元的乘法群（特别地，剩余类环的乘法群）。我们给出了该方案的两种版本。