Simulation of a Homomorphic Encryption System

Cryptology is defined as the science of making communication incomprehensible to third parties who have no right to read and understand the data or messages. Cryptology is divided into cryptography, which is the science of securing data, and cryptanalysis, which is the science of analyzing and breaking secure communication. The main terms in cryptology are given in Table 1.

The goal of encryption is to ensure confidentiality of data in communication and storage processes. Homomorphic encryption is a form of encryption that allows specific types of computations like addition or multiplication to be carried out on ciphertext. The encrypted result will be the same when decryption is done. Widespread use of cloud computing raises the question of whether it is possible to delegate the processing of data without giving access to it. Encrypting one’s data with a conventional encryption scheme to protect one’s privacy seems to undermine the benefits of cloud computing since it is impossible to process the data without the decryption key [1]. However, in homomorphic encryption; only the encrypted version of the data is given to the untrusted computer to process. The computer will perform the computation on this encrypted data, without knowing anything on its real value. Finally, it will send back the result, and whoever has the proper deciphering key can decrypt the cryptogram correctly. For coherence, the decrypted result will be equal to the intended computed value.

Homomorphic encryption schemes are methods that allow the transformation of ciphertexts C(M) of message M, to ciphertexts C(f(M)) of a computation/function of message M, without disclosing the message. Generally, an encryption scheme contains a three-step algorithm. They are

Key Generation—creates two keys, i.e. the secret key sk and the public key pk.

In homomorphic encryption, the encryption of the product of two numbers is equal to the product of the encryptions of the numbers: E(a.b)=E(a).E(b)E(a.b) = E\left( {a).E(b} \right)E is an encryption algorithm or it is also known as a scheme. Schemes can be thought as Somewhat Homomorphic and Fully Homomorphic. Fully Homomorphic: Any kind of operations on ciphertexts can be done. This idea came from Rivest in 1978 but Craig Gentry was the first one whose scheme works on arbitrary functions. Somewhat Homomorphic: In this type, two ciphertexts can be added or a ciphertext and a plaintext can be multiplied [6] .

In this paper, SWHE is examined. The length of the plaintext and encryption time comparison has been made. To do that, Gentry’s encryption scheme is used [7] . The parameters will be ρ=λ, ρ′=2λ, η=O˜(λ2), λ=O˜(λ5)\rho = \lambda ,{\text{ }}\rho ' = 2\lambda ,{\text{ }}\eta = {\text{\tilde O}}({\lambda ^2}),{\text{ }}\lambda = {\text{\tilde O}}({\lambda ^5})λ is a safety parameter. Security parameters associated with the security of the scheme, usually take dozens to hundreds of bits.

Key generation: We choose η bit odd prime numbers p and θ bits odd prime numbers q randomly and order N=pq. Then choose two random integers l ∈ [0,2γ/p] h∈ [−2p, 2p], and calculate x=pl+2h.

Set public key pk=(N,x)=, private key sk=p.

Encryption (pk,m): Given a plaintext m∈ {0,1}*; we choose two random integers r1∈ (−2p,2p) and r2∈ (−2p,2p). According to the public key pk=(N,x), we can calculate c=m+2r1+ r2x mod N (c is the result of ciphertext.)c = m + 2{r_1} + {r_2}x\,\,\,\,\,\bmod \,\,\,\,N\,\,\,\left( {c\,\,\,{\text{is}}\,\,\,{\text{the}}\,\,\,{\text{result}}\,\,\,{\text{of}}\,\,\,{\text{ciphertext}}.} \right)\Decryption (sk,c): According to the given ciphertext, make use of private key sk to calculate: m′=(c mod p)mod2m' = (c{\text{ mod }}p){\text{mod}}2Evaluate (pk, C, c1,c2,ct): Given a Boolean circuit C with t inputs and t ciphertexts ci. Let us put T ciphertext into extended circuits to perform all its operations, then verify the result of the circuit’s output in (1) to see if it conform (2) [7]. (1)c*=evaluate(pk,C,c)c* = {\text{evaluate}}(pk,C,c)(2)Dec(sk,c*)=C(m1,m2,…,mt)Dec(sk,c*) = C({m_1},{m_2}, \ldots ,{m_t})

In the experimental environment, safe parameter λ is set to the length of the 10 bits and the order of magnitude as 10⁵. So the order of magnitude for p = λ as 10⁵. p′ = 2λ as 10⁵, the order of magnitude for η = O(λ²) as 10¹⁰, the order of magnitude for θ = O(λ⁴) as 10²⁰, and the order of magnitude for γ=~ O (λ⁵) as 10²⁵. Algorithm keygen of the data is determined by the above parameters. The relationship between the plaintext and ciphertext size is shown in Figure 1. The simulation time is calculated in seconds. It is observed that when plaintext size is increased, the ciphertext size is also increased. But the growthe rate is not high. Still, it is gradually increased.

Fig. 1

Cryptography is a powerful tool to protect information. In recent years, cryptography and cryptanalysis had been improved. Widespread use of cloud computing raises the question of whether it is possible to delegate the processing of data without giving access to it. Homomorphic encryption is a new way to protect private data. Because it allows making computation without decrypting data. It is a new field and research is going on.