In this paper,
we have proposed new variants of
the El-Gamal and
the Cramer-Shoup encryption schemes.
In our schemes,
the anonymity property holds
even if each user chooses an arbitrary prime q
where |q| = k and p = 2q+1 is also prime.
More precisely,
our El-Gamal variants
provide anonymity
against the chosen-plaintext attack, and
our Cramer-Shoup variants
provide anonymity
against the adaptive chosen-ciphertext attack.
These anonymity properties are proved
under a slightly weaker assumption than
the DDH assumption.
Furthermore,
our El-Gamal variants are secure in the sense of IND-CPA, and
our Cramer-Shoup variants are secure in the sense of IND-CCA2.