Commitment scheme is a basic component of many cryptographic protocols, such as coin-tossing, identification schemes, zero-knowledge and multi-party computation. In order to prevent man-in-middle attacks, non-malleability is taken into account. Many forming works focus on designing non-malleable commitments schemes based on number theory assumptions. In this paper we give a general framework to construct non- interactive and non-malleable commitment scheme with respect to opening based on more general assumptions called q-one way group homomorphisms (q-OWGH). Our scheme is more general since many existing commitment schemes can be deduced from our scheme. Commitment scheme is a basic component of many cryptographic protocols, such as coin-tossing, identification schemes, zero-knowledge and multi-party computation. In order to prevent man-in-middle attacks, non-malleability is taken into account. Many forming works focus on designing non-malleable commitments schemes based on number theory assumptions. In this paper we give a general framework to construct non-interactive and non-malleable commitment scheme with respect to opening to based on more general analogues of q-one way group homomorphisms (q-OWGH). Our scheme is more general since many existing commitment schemes can be deducted from our scheme.