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A decade after Gentry\u2019s breakthrough discovery of how we might support arbitrary computations on encrypted data, many studies followed and improved various aspects of HE, such as faster bootstrapping and ciphertext packing. However, the topic of how to support secure computations on ciphertexts encrypted under multiple keys does not receive enough attention. This capability is crucial in many application scenarios where data owners want to engage in joint computations and are preferred to protect their sensitive data under their own secret keys. Enabling this capability is a non-trivial task. In this article, we present a comprehensive survey of the state-of-the-art multi-key techniques and schemes that target different systems and threat models. In particular, we review recent constructions based on Threshold Homomorphic Encryption (ThHE) and Multi-Key Homomorphic Encryption (MKHE). We analyze these cryptographic techniques and schemes based on a new secure outsourced computation model and examine their complexities. We share lessons learned and draw observations for designing better schemes with reduced overheads.<\/jats:p>","DOI":"10.1145\/3477139","type":"journal-article","created":{"date-parts":[[2021,10,8]],"date-time":"2021-10-08T22:02:57Z","timestamp":1633730577000},"page":"1-37","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":27,"title":["Computing Blindfolded on Data Homomorphically Encrypted under Multiple Keys: A Survey"],"prefix":"10.1145","volume":"54","author":[{"given":"Asma","family":"Aloufi","sequence":"first","affiliation":[{"name":"Taif University, Al Hawiyah, Taif, Saudi Arabia"}]},{"given":"Peizhao","family":"Hu","sequence":"additional","affiliation":[{"name":"Rochester Institute of Technology, Rochester, NY"}]},{"given":"Yongsoo","family":"Song","sequence":"additional","affiliation":[{"name":"Seoul National University, Gwanak-gu, Seoul, Republic of Korea"}]},{"given":"Kristin","family":"Lauter","sequence":"additional","affiliation":[{"name":"Microsoft Research, Redmond, WA"}]}],"member":"320","published-online":{"date-parts":[[2021,10,8]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/3214303"},{"key":"e_1_2_1_2_1","volume-title":"the International Workshop on Privacy Engineering (IWPE\u201919)","author":"Aloufi Asma","year":"2019","unstructured":"Asma Aloufi and Peizhao Hu . 2019 . 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