Task1: Investigate the details of the homomorphic properties of public-key encry

Question

Task1: Investigate the details of the homomorphic properties of public-key encryptions. This investigation has to be carried out with the following outcomes in mind: i) What is homomorphism in cryptosystems and what are the different homomorphic encryption properties? Ii) What is the difference between a partially homomorphic cryptosystem and a fully homomorphic cryptosystem? iii) Details of the homomorphic properties exhibited by popular public-key cryptosystems such as RSA, Elgamal, Paillier, Goldwasser-Micali, etc. iv) Example of applications in which such a property of public-key cryptosystem would be useful. Task2: In this task, you will verify the homomorphic properties of at least two public-key cryptosystems by means of an implementation. You do not need to implement the actual cryptosystems as implementations of popular public-key cryptosystems such as RSA, Elgamal, etc., are freely available in the GNU libgcrypt library (http://www.gnu.org/software/libgcrypt/) or the crypto library (http://www.openssl.org/docs/crypto/crypto.html).

Explanation / Answer

The homomorphic

feature referred to is a minimalistic one; that is, the length of a homomorphically generated

encryption should be independent of the number of ciphertexts from which it was created. We

do not require anything else on the distribution of homomorphically generated encryptions (in

particular, we do not require them to be distributed like real ciphertexts). Our resulting publickey

scheme is homomorphic in the following sense. If i+1 repeated applications of homomorphic

operations can be applied to the private-key scheme, then i repeated applications can be applied

to the public-key scheme.

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