The DGHV (Dijk-Gentry-Halevi-Vaikutanathan) which is an asymmetric encryption scheme based on the computation over real integers. The basic DGHV scheme provides the homomorphic behavior for a limited circuit depth. Bootstrapping is a technique introduced the literature to make the scheme fully homomorphic and supports unlimited circuit depth.  Basic symmetric scheme

In 2007, Yuval et al. have expanded the set of branching programs which are directed acyclic graphs in such a way that every node has two outgoing edges with labeled binary 0 and 1. Furthermore, they proposed a public key encryption scheme by evaluating the branching programs on the encrypted data. In addition, Melchor et al. proposed a generic construction method to obtain a chained encryption scheme that allowing the homomorphic evaluation of constant depth circuit over ciphertext. The chained encryption scheme is obtained from well-known encryption schemes with some homomorphic properties. For example, they showed how to obtain a combination of BGN and KTX (kawachi et. al. suggested an additively homomorphic encryption scheme over a large cyclic group, which depends on the hardness of underlying lattice problems.). Hence, the resulting combined scheme allows arbitrary additions and two multiplicati-ons.

One of the most significant steps toward an FHE scheme was introduced by Boneh-Goh-Nissim (BGN) in [Boneh et al. 2005]. BGN evaluates 2-DNF5 formulas on ciphertext and it supports an arbitrary number of additions and one multiplication by keeping the ciphertext size constant. The hardness of the scheme is based on the subgroup decision problem. Subgroup decision problem simply decides whether an element is a member of a subgroup Gp of group G of Composite order n = pq, where p and q are distinct primes.

SYY describes first SWHE scheme over a semi-group. It is another idea of evaluating operations on encrypted data that is realized over different sets. This idea requires fewer properties than a group. This scheme Polynomially supports many ANDing of ciphertexts with one OR/NOT gate. Therefore, the ciphertext size increases due to the constant multiplication with each OR/NOT gate evaluation. However, the limits of the circuit depth evaluation are increased.

Polly Cracker scheme is one of the first SWHE that applies two operations, i.e., multiplication and addition over the ciphertexts. Background By the term Polly Cracker-type cryptosystem, we mean a family of cryptosystems starting from the early 1990s that propose to base their security on the difficulty of computing Grobner bases. In its public key version and the most simple form, the public key is an ideal I in a polynomial ring (given by sufficiently many polynomials of degree b from I) and the secret key is a Grobner basis for I consisting of polynomials of degree d≤ b. let I be some ideal in P := F[x₀, . . . , x_n−1]. Denote an injective function mapping bit strings to elements in the quotient ring P/I by Encode (·), and its inverse by Decode (·). If Decode (Encode (m0) ◦ Encode (m1)) = m0 ◦ m1 for ◦ ∈ {+, ·}

we can encrypt a message m as c = f + Encode (m) for f randomly chosen in I.

In Partially Homomorphic encryption, we can perform either multiplication or addition that means both addition and multiplication cannot be performed simultaneously. But Somewhat Homomorphic encryption allows us to perform both operations at the same time. There are useful SWHE examples [Yao 1982; Sander et al. 1999; Boneh et al. 2005; Ishai and Paskin 2007] in the literature before 2009. There are several Somewhat Homomorphic Encryption (SWHE) are globally accepted. One of the popular methods is BGN encryption scheme. This scheme was developed by Beneh-Goh-Nissimb at Stanford University, which is also the first practical Somewhat Homomorphic Encryption. Before this encryption scheme, all the operations are limited to addition or multiplication only. This scheme gave a major advancement towards the practical Fully Homomorphic Encryption. This scheme gave a major advancement towards the practical Fully Homomorphic Encryption.

The ElGamal encryption scheme is a public-key encryptionalgorithm based on the Diffie–Hellman key exchange. It was invented by Taher Elgamal in 1985. The ElGamal encryption scheme can be defined over any cyclic group G. Its security depends upon the difficulty of a certain problem in G related to computing discrete logarithms. The ElGamal encryption scheme consists of three components: the key generation, the encryption algorithm, and the decryption algorithm.

Goldwasser–Micali Encryption Scheme: The Goldwasser–Micali (GM) encryption scheme [isa public-key encryption algorithm developed by Shafi Goldwasser and Silvio Micali in 1982. GM has the distinction of being the first probabilistic public-key encryption scheme which is provably secure under standard cryptographic assumptions. However, it is not an efficient cryptosystem, as ciphertexts may be several hundred times larger than the initial plaintext. GM consists of three algorithms: a probabilistic key generation algorithm which produces a public and a private key, a probabilistic encryption algorithm, and a deterministic decryption algorithm.

RSA (Rivest Shamir Adleman): RSA is an early example of PHE and introduced by Rivest, Shamir, and Adleman [Rivest et al. 1978b] shortly after the invention of public key cryptography by Diffie Helman [Diffie and Hellman 1976]. RSA is the first feasible achievement of the public key cryptosystem. Moreover, the homomorphic property of RSA was shown by Rivest, Adleman, and Dertouzous [Rivest et al. 1978a] just after the seminal work of RSA. Indeed, the first attested use of the term “privacy homomorphism” is introduced in [Rivest et al. 1978a]. The security of the RSA cryptosystem is based on the hardness of factoring problem of the product of two large prime numbers.

There are several useful PHE examples [Rivestet al. 1978b; Goldwasser and Micali 1982 ElGamal1985; Benaloh 1994; Naccache and Stern 1998; Okamoto and Uchiyama 1998; Paillier 1999; Damgård and Jurik 2001; Kawachi et al. 2007] in the literature. Each has improved the PHE in some way. However, in this section, we primarily focus on major PHE schemes that are the basis for many other PHE schemes.