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A Survey Report on Partially Homomorphic Encryption Techniques in Cloud Computing International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 2 Issue 12, December - 2013 A Survey Report On Partially Homomorphic Encryption Techniques In Cloud Computing 1. S. Ramachandram 2. R.Sridevi 3. P. Srivani 1. Professor &Vice Principal, Oucollege Of Engineering,Dept Of Cse,Hyd 2. Associate Professor, Jntuh, Dept Of Cse,Hyd 3. Research Scholar & Asso.Prof., Det Of Cse, Hitam, Hyd Abstract: Homomorphic encryption is a used to create secure voting systems, form of encryption which allows specific collision-resistant hash functions, private types of computations to be carried out on information retrieval schemes and enable cipher text and obtain an encrypted result widespread use of cloud computing by which decrypted matches the result of ensuring the confidentiality of processed operations performed on the plaintext. As data. cloud computing provides different services, homomorphic encryption techniques can be There are several efficient, partially used to achieve security. In this paper , We homomorphic cryptosystems. Although a presented the partially homomorphic cryptosystem which is unintentionally encryption techniques. homomorphic can be subject to attacks on this basis, if treated carefully, Keywords: Homomorphic, ElGamal, Pailler, homomorphism can also be used to perform RSA, Goldwasser–Micali , Benaloh, computations securely. Section 2 describes Okamoto Uchiyama cryptosystemIJERT, IJERTabout the partially homomorphic encryption Naccache–Stern cryptosystem, RNS. techniques. 2. Partially Homomorphic Encryption Techniques 1. INTRODUCTION RSA Security is a desirable feature in modern system architectures. Homomorphic In cryptography, RSA[1] is an asymmetric encryption would allow the chaining encryption system. If the RSA public key is together of different services without modulus and exponent , then the exposing the data to each of those services. encryption of a message is given by For example a chain of different services . from different companies could 1) calculate the tax 2) the currency exchange rate 3) The homomorphic property is then shipping, on a transaction without exposing the unencrypted data to each of those services. Homomorphic encryption schemes are malleable by design. The homomorphic property of various cryptosystems can be IJERTV2IS120681 www.ijert.org 3278 International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 2 Issue 12, December - 2013 ElGamal Encryption Bob calculates the shared secret . In cryptography, the ElGamal encryption system[2] is an asymmetric encryption Bob converts his secret message into algorithm for public key cryptography an element of . which is based on the Diffie Helman Bob calculates . exchange . It was described by Taher Bob sends the cipher text ElGamal in 1984. ElGamal encryption is used in the free GNU privacy Guard to Alice. software, recent versions of PGP, and other cryptosystems. The Digital Signature Note that one can easily find if one Algorithm is a variant of ElGamal Signature knows . Therefore, a new is generated Algorithm, which should not be confused for every message to improve security. For with ElGamal encryption. this reason, is also called an ephemeral key. ElGamal encryption can be defined over any cyclic Group . Its security depends upon Decryption the difficulty of a certain problem in related to computing discrete logarithms. The decryption algorithm works as follows: to decrypt a cipher ext with her Key Generation private key , The key generator works as follows: Alice calculates the shared secret Alice generates an efficient description of a multiplicative cyclicIJERT IJERT and then computes group of order with generator which she then converts back into . the plaintext message , where Alice chooses a random from is inverse of In the group . (E.g. modular multiplicative inverse if is a subgroup of a Alice computes . multiplication group of integers Alice publishes , along with the modulo n). description of , as her public key. Alice retains as her private The decryption algorithm produces the key which must be kept secret. intended message, since Encryption The encryption algorithm works as follows: Efficiency to encrypt a message to Alice under her public key , ElGamal encryption is probabilistic meaning that a single plain text can be Bob chooses a random from encrypted to many possible cipher texts, , then calculates with the consequence that a general . ElGamal encryption produces a 2:1 IJERTV2IS120681 www.ijert.org 3279 International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 2 Issue 12, December - 2013 expansion in size from plaintext to cipher Goldwasser–Micali text. The Goldwasser–Micali (GM) crypto Encryption under ElGamal requires two system[3] is an asymmetric key encryption exponentiation. However, these algorithm developed by Shaff Goldwasser exponentiations are independent of the and Silvio Micali in 1982. GM has the message and can be computed ahead of time if need be. Decryption only requires one distinction of being the first probabilistic exponentiation. public-key encryption scheme which is provably secure under standard The division by can be avoided by using cryptographic assumptions. However, it is an alternative method for decryption. To not an efficient cryptosystem, as cipher texts decrypt a cipher text with Alice's may be several hundred times larger than the private key , initial plaintext. To prove the security properties of the cryptosystem, Goldwasser Alice calculates and Micali proposed the widely used . definition of semantic security. is the inverse of . This is a consequence Because encryption is performed using a of Langranges Theorem, because probabilistic algorithm, a given plaintext may produce very different cipher texts each . time it is encrypted. This has significant advantages, as it prevents an adversary from recognizing intercepted messages by Alice then computes , comparing them to a dictionary of known which she then converts back into IJERT IJERTcipher texts. the plaintext message . The scheme relies on deciding whether a The decryption algorithm produces the given value x is a square mod N, given the intended message, since factorization (p, q) of N. This can be accomplished using the following . procedure: In the ElGamal cryptosystem, in a 1. Compute xp = x mod p, xq = x mod q. group , if the public key is , where , and 2. If and is the secret key, then the , then x encryption of a message is is a quadratic residue mod N. , for some random . The Key Generation homomorphic property is then Alice generates two distinct large prime numbers p and q, randomly and independently of each other. IJERTV2IS120681 www.ijert.org 3280 International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 2 Issue 12, December - 2013 1. Alice computes N = p q. quadratic non-residue x, then the encryption 2. She then finds some non-residue x of a bit b is , for such that the legendra symbols some random . The satisfy and hence homomorphic property is then the Jacobi symbol is +1. The value x can for example be found by selecting random values and testing where denotes addition modulo 2. the two Legendre symbols. If (p, q) = 3 mod 4 (i.e., N is a Blum integer), Benaloh Cryptosystem then the value N − 1 is guaranteed to have the required property. The Benaloh Cryptosystem[4] is an extension of the Goldwasser micali The public key consists of (x, N). The secret cryptosystem(GM) created in 1994 by Josh key is the factorization (p, q). (Cohen) Benaloh. The main improvement of the Benaloh Cryptosystem over GM is that Message encryption longer blocks of data can be encrypted at once, whereas in GM each bit is encrypted Suppose Bob wishes to send a message m to individually. Like many public key Alice: cryptosystem, this scheme works in the 1. Bob first encodes m as a string of group where n is a product of bits (m1, ., mn). two large primes. This scheme is 2. For every bit , Bob generates a homomorphic and hence malleable. random value from the group of units modulo N, or . Key Generation He outputs the valueIJERT IJERT A public/private key pair is generated as . follows: Bob sends the cipher text (c1, .. , cn). Choose a block size r. Choose large primes p and q such Message Decryption that r divides (p-1), gcd(r, (p-1)/r) = 1 and gcd(q-1,r) = 1. Alice receives (c1, ..., cn). She can recover m Set n = pq using the following procedure: Choose such that 1. For each i, using the prime . factorization (p, q), Alice determines whether the value ci is a quadratic The public key is then y, n, and the private residue; if so, mi = 0, otherwise mi = key is the two primes p, q. 1. Message encryption Alice outputs the message m = (m1, ..., mn). To encrypt a message m, where m is taken to In the Goldwasser micali cryptosystem , if be an element in the public key is the modulus m and IJERTV2IS120681 www.ijert.org 3281 International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 2 Issue 12, December - 2013 Choose a random Pailler cryptosystem Set The Paillier crypto system[5], named after Message decryption and invented by pascal pailler in 1999, is a probabilistic asymmetric algorithm for public key cryptography. The problem of computing n-th residue classes is believed to To understand decryption, we first notice be computationally difficult. The decisional that for any m, u we have composite residuosity assumption is the intractability hypothesis upon which this Since m < r and cryptosystem is based. , we can conclude that The scheme is an additive homomorphic if and cryptosystem; this means that, given only only if m = 0. So if is the public-key and the encryption of an encryption of m, given the secret key p, q and , one can compute the encryption of we can determine whether m=0. If r is small, . we can decrypt z by doing an exhaustive -I search, i.e. decrypting the messages y z for i Key Generation from 1 to r.
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