Simultaneous Approximation for Lattice-Based Cryptography

We define two new problems called SIAP and CAP related to solving SIVP and CVP in a subset of lattices called Simultaneous Approximation (SA) lattices. We give dimension- and gap-preserving, deterministic polynomial-time and space reductions from SVP${}_{\gamma }$, SIVP${}_{\gamma }$, and CVP${}_{\gamma }$ to their corresponding problems in SA lattices. These reductions show that instances of these problems in SA lattices are just as hard as general instances and thus are interesting new problems to consider for use in cryptography. We also show that the reductions are optimal in regards to integer inflation.