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Top1. Introduction
Processing encrypted data in the cloud computing is today possible when data are encrypted with a fully homomorphic encryption scheme. This category of encryption has goal to allow computations in the ciphertext space without decrypting and without revealing the decryption key. Given its importance in cloud security and many other applications (El-yahyaoui & Ech-cherif El Kettani, 2017), (Armknecht, et al., 2015), (Damgård, Groth, & Salomonsen, 2002), it was early conjectured by Rivest et al. (Rivest, Adleman, & Dertouzos, 1978) under the name of privacy homomorphism. Gentry (Gentry, 2009) who gives it the famous name of fully homomorphic encryption solved this conjecture in 2009 after a breakthrough thesis work.
Simplification and improvement of fully homomorphic encryption schemes is the current occupation of many cryptographers. The majority of succeeding works (Smart & Vercauteren, 2009), (van Dijk, Gentry, Halevi, & Vaikuntanathan, 2009), (Vikuntanathan & Brakerski, 2011), (Brakerski, 2012), (Gentry & Halevi, 2011) after Gentry’s breakthrough has as paramount objective to make simpler the design and ameliorate performances of fully homomorphic encryption algorithms. One of the significant ameliorations consists in adding new capacities to this class of algorithms (Gentry, Sahai, & Waters, 2013), (El-yahyaoui & Ech-cherif El kettani, 2017), (Asharov, et al., 2012), (Lopez-Alt, Tromer, & Vaikuntanathan, 2012). Verification capacity is one of the best characteristics that can hold a fully homomorphic encryption scheme.
A verifiable encryption scheme is a cryptosystem that permits us to prove some properties about a hidden value in a ciphertext without decrypting it. If the verification option is integrated with homomorphic capacities in the same encryption scheme, it becomes a verifiable fully homomorphic encryption scheme (VFHE). Consequently, a VFHE scheme is a particular case of FHE schemes for which the capacities of computing over encrypted data and delegating calculations on confidential data, to a remote cloud server, are given to the cloud client with the possibility of verifying the rightness of its outsourced computations.
In practice, FHE and VFHE schemes are costing. It generates a huge amount of noise and becomes ugly when evaluating multiplications on ciphertexts because the noise growth is square in general. As long as we add capacities to fully homomorphic encryption schemes (El-yahyaoui & Ech-cherif El kettani, 2017), (Gentry, Sahai, & Waters, 2013), (Lopez-Alt, Tromer, & Vaikuntanathan, 2012), it becomes uglier and costlier in practice.
Improvement of FHE has several techniques and tools. Each technique has its special impact on client or server side. Techniques that are recommended to client-side improvement allow enhancing the encryption runtime and take into consideration client processing powers. While server-side improvement techniques permit to reduce the client consumption in terms of processing and space storage in the cloud. This reduction minimizes the costumer’s billing of cloud resources consumption.