## Homomorphic Signature Schemes - A Survey

**Book Title**

Homomorphic Signature Schemes - A Survey

**Authors**

Giulia Traverso, Denise Demirel, Johannes Buchmann (Technische Universität Darmstadt)

**Preface**

In the last years, there has been an increasing interest in homomorphic signature schemes. Thus, many schemes have been proposed that are suitable for a lot of different applications. In this work, we overcome the extensive state of the art by presenting a survey of the existing approaches and the properties they provide. In addition, we look into three interesting use cases for homomorphic operations on authenticated data; these are electronic voting, smart grids, and electronic health records. We discuss their requirements, show to what extent the existing solutions meet these conditions, and highlight promising directions for future work. Homomorphic signature schemes have been initially designed to establish authentication in network coding and to address pollution attacks (see [18]). However, since they allow for computations on authenticated data, they are also a useful primitive for many other applications. In fact, after Johnson et al. introduced a formal definition and a precise framework for homomorphic signatures in 2002 (see [46]), in the following years, many schemes have been presented and discussed. The first schemes proposed only allow to perform linear computations on authenticated data (e.g., [71, 72, 76], and [24]). These approaches have been further improved with respect to efficiency, security, and privacy [5–7, 15, 18, 21, 22, 22, 34, 36, 65]. In addition, to be more flexible, solutions have been developed supporting polynomial functions [14, 23, 43], or even coming without any restrictions on the functions themselves, so-called fully homomorphic signature schemes [19, 41]. However, all these solutions assume that each input signature has been generated using the same private key. To overcome this restriction, the homomorphic property has been added to the aggregate signature schemes [45, 74] allowing for operations on signatures generated using even different secret-public key pairs. In this work, we start by providing a formal definition of these four types of homomorphic signature schemes. First, the passage from the digital signature schemes to the homomorphic ones is formally described, where the novelties introduced by the homomorphic property itself are highlighted. Afterward, it is described how to obtain the linearly homomorphic signature schemes from the merely homomorphic ones. And then, starting from the linearly homomorphic signature schemes, it is shown how to derive the schemes supporting polynomial v vi Preface functions and how to define the fully homomorphic signature schemes. Finally, schemes that allow computations on signatures generated using different secretpublic key pairs are formally described. Up to our knowledge, this survey is the first such work providing both a description of each single homomorphic signature scheme and a description of the whole general framework in a methodical and didactic approach. Indeed the survey proposed in [73] is not up to date, while in [20] the existing homomorphic signature schemes are just listed, without any deeper discussions. Furthermore, in this survey, we also discuss the possible use cases electronic voting, smart grids, and electronic health records. For each use case, concrete examples of how improvements can be achieved by the usage of homomorphic signature schemes are provided, together with the definition of the minimal requirements these schemes should fulfill. Furthermore, it is shown which of the currently existing homomorphic signature schemes are suitable for which of the use cases in question. When that is not the case, directions for future works are proposed. In Chap. 1, the definition of general digital signature schemes is recalled, and the formal description of the homomorphic signature schemes is provided. Chapter 2 provides a description of the linearly homomorphic signature schemes, the homomorphic signature schemes for polynomial functions, the fully homomorphic signature schemes, and the homomorphic aggregate signature schemes. In Chap. 3, interesting properties of homomorphic signature schemes are discussed. The description of each of the currently existing homomorphic signature scheme and the properties they provide follow in Chap. 4. In Chap. 5, the usage of homomorphic signature schemes for each of the aforementioned use cases is presented. Finally, in Chap. 6, a conclusion is given and possible directions for future work are shown.

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