Group homomorphic encryption

2012 – Armknecht, Katzenbeisser – Group homomorphic encryption

https://core.ac.uk/reader/11489070

Non-formal, the homomorphic encryption property means that adding / multiplying two ciphertexts results the encryption of two added / multiplied plaintexts.
So we can talk about additive homomorphism and multiplicative homomorphism.

1. quotes from the article Additively Homomorphic IBE from Higher Residuosity

“GHE only permits evaluation of a single algebraic operation” (as it is said here)

“1. It is used as a building block in protocols for Private Information Retrieval[5], Electronic Voting [6–10], Oblivious Polynomial Evaluation [11], Private Outsourced Computation [12] and the Millionaire’s Problem [13].

2. Fully Homomorphic Encryption (FHE) is currently impractical for many applications, and even if it were to become more practical, it may add unnecessary overhead, especially in applications that only require a single algebraic operation” (see this article)

“GHE is the “classical” flavor of homomorphic encryption. It allows unbounded applications of the group operation. Goldwasser and Micali [14] constructed the first GHE scheme. The Goldwasser-Micali (GM) cryptosystem supports addition modulo 2 i.e. the XOR operation. Other additively-homomorphic GHE schemes from the literature include Benaloh [6], Naccache-Stern [15], Okamoto-Uchiyama[16], Paillier [17] and Damg ̊ard-Jurik [10]. Other instances of GHE include [18–20].

Existing identity-based GHE (IBGHE) schemes such as those based on pairings are typically multiplicatively homomorphic. It is a well-known that a scheme with a multiplicative homomorphism can be transformed into one with an additive homomorphism, where the addition takes place in the exponent, and a discrete logarithm problem must be solved to recover the result. In this case, we usually get a bounded (aka “quasi”) additively homomorphic scheme, but it is not group homomorphic in the sense of the definition considered in this paper since one cannot perform an unbounded number of homomorphic operations. However, to the best of our knowledge, the only existing “pure” (i.e. supporting modular addition) additively-homomorphic instance of IBGHE in the literature is the variant of the Cocks scheme due to Clear, Hughes and Tewari [21] that is XOR-homomorphic i.e. it supports addition modulo 2. Applications of IBGHE are explored in [21] but can be extended to private information retrieval (PIR)[22] (instantiating the protocol from [5] with an IBGHE scheme instead of a public-key GHE scheme), data aggregation in wireless sensor networks (IBE has been applied to wireless sensor networks already in [23–26]) and participatory sensing (G ̈unther et al. [27] use additively homomorphic IBE for data aggregation in a participatory sensing system).”