The **Damgård–Jurik cryptosystem** is a generalization of the Paillier cryptosystem. It uses computations modulo where is an RSA modulus and a (positive) natural number. Paillier's scheme is the special case with . The order (Euler's totient function) of can be divided by . Moreover can be written as the direct product of . is cyclic and of order, while is isomorphic to . For encryption, the message is transformed into the corresponding coset of the factor group and the security of the scheme relies on the difficulty of distinguishing random elements in different cosets of . It is semantically secure if it is hard to decide if two given elements are in the same coset. Like Paillier, the security of Damgård–Jurik can be proven under the decisional composite residuosity assumption.

Read more about Damgård–Jurik Cryptosystem: Key Generation, Encryption, Decryption, Simplification

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**Damgård–Jurik Cryptosystem**- Simplification

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**Damgård–Jurik**can be simplified in the following way The base g is fixed as ...