Homomorphic encryption

Homomorphic encryption is a form of encryption that allows computations to be carried out on ciphertext, thus generating an encrypted result which, when decrypted, matches the result of operations performed on the plaintext.

This is sometimes a desirable feature in modern communication system architectures. Homomorphic encryption would allow the chaining together of different services without exposing the data to each of those services. For example, a chain of different services from different companies could calculate 1) the tax 2) the currency exchange rate 3) shipping, on a transaction without exposing the unencrypted data to each of those services.Homomorphic encryption schemes are malleable by design. This enables their use in cloud computing environment for ensuring the confidentiality of processed data. In addition the homomorphic property of various cryptosystems can be used to create many other secure systems, for example secure voting systems, collision-resistant hash functions, private information retrieval schemes, and many more.

There are several partially homomorphic crypto-systems, and also a number of fully homomorphic crypto-systems. Although a crypto-system which is unintentionally malleable can be subject to attacks on this basis, if treated carefully homomorphism can also be used to perform computations securely.

In the following examples, the notation ${\displaystyle {\mathcal {E}}(x)}$ is used to denote the encryption of the message x.

If the RSA public key is modulus ${\displaystyle m}$ and exponent ${\displaystyle e}$, then the encryption of a message ${\displaystyle x}$ is given by ${\displaystyle {\mathcal {E}}(x)=x^{e}\;{\bmod {\;}}m}$. The homomorphic property is then

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