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In this work, we study a new multivariate quadratic (MQ) assumption that can be used to construct public-key encryptions. In particular, we research in the following two directions: We establish a precise asymptotic formulation of a family of hard MQ problems, and provide empirical evidence to confirm the hardness. We construct public-key encryption schemes, and prove their security under the hardness assumption of this family. Also, we provide a new perspective to look at MQ systems that plays a key role to our design and proof of security. As a consequence, we construct the first public-key encryption scheme that is provably secure under the MQ assumption. Moreover, our public-key encryption scheme is efficient in the sense that it only needs a ciphertext length L +poly(k ) to encrypt a message M ∈{0, 1}L for any un-prespecified polynomial L , where k is the security parameter. This is essentially optimal since an additive overhead is the best we can hope for.