The present work builds on previous investigations of the authors (and their collaborators) regarding bridges, a certain type of morphisms between encryption schemes, making a step forward in developing a (category theory) language for studying relations between encryption schemes. Here we analyse the conditions under which bridges can be performed sequentially, formalizing the notion of composability. One of our results gives a sufficient condition for a pair of bridges to be composable. We illustrate that composing two bridges, each independently satisfying a previously established IND-CPA security definition, can actually lead to an insecure bridge. Our main result gives a sufficient condition that a pair of secure composable bridges should satisfy in order for their composition to be a secure bridge. We also introduce the concept of a complete bridge and show that it is connected to the notion of Fully composable Homomorphic Encryption (FcHE), recently considered by Micciancio. Moreover, we show that a result of Micciancio which gives a construction of FcHE schemes can be phrased in the language of complete bridges, where his insights can be formalised in a greater generality.

翻译：本文基于作者（及其合作者）先前关于桥（加密方案之间的一种特定态射）的研究，在发展用于研究加密方案间关系的（范畴论）语言方面迈出了一步。本文分析了桥可以顺序执行的条件，将可组合性的概念形式化。研究结果之一给出了桥对可组合的充分条件。我们证明，组合两个各自独立满足先前已建立的IND-CPA安全定义的桥，实际上可能导致不安全的桥。主要结果给出了安全可组合桥对应满足的充分条件，以确保其组合后的桥是安全的。我们还引入了完全桥的概念，并证明其与Micciancio近期研究的完全可组合同态加密（FcHE）概念相关。此外，我们表明Micciancio关于FcHE方案构造的结果可以用完全桥的语言表述，其洞见可在更泛化的框架下形式化。