Damgård–Jurik cryptosystem

The Damgård–Jurik cryptosystem[1] 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.

Key generation

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  1. Choose two large prime numbers p and q randomly and independently of each other.
  2. Compute   and  .
  3. Choose an element   such that   for a known   relative prime to   and  .
  4. Using the Chinese Remainder Theorem, choose   such that   and  . For instance   could be   as in Paillier's original scheme.
  • The public (encryption) key is  .
  • The private (decryption) key is  .

Encryption

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  1. Let   be a message to be encrypted where  .
  2. Select random   where  .
  3. Compute ciphertext as:  .

Decryption

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  1. Ciphertext  
  2. Compute  . If c is a valid ciphertext then  .
  3. Apply a recursive version of the Paillier decryption mechanism to obtain  . As   is known, it is possible to compute  .

Simplification

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At the cost of no longer containing the classical Paillier cryptosystem as an instance, Damgård–Jurik can be simplified in the following way:

  • The base g is fixed as  .
  • The decryption exponent d is computed such that   and  .

In this case decryption produces  . Using recursive Paillier decryption this gives us directly the plaintext m.

See also

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References

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