Public-Key Cryptosystems from the Worst-Case Shortest Vector Problem.

We construct public-key cryptosystems that are secure assuming the worst-case hardness of approximating the minimum distance oil n-dimensional lattices to wit, hill small poly(n) factors. Prior cryptosystems with worst-case connections were based either on the shortest, vector problem for a special class of lattices (Ajtai and Dwork. STOC 1997; Regev; J. ACM 2004), or on the conjectured hardness of lattice problems for quantum algorithms (Regev, STOC 2005). Our main technical innovation is a reduction from variants of the shortest vector problem to corresponding versions, of the "learning with errors" (LWE) problem; previously, only a quantum reduction of this kind was known. As an additional contribution, we construct a natural chosen cryptosystem having a much simpler description and tighter underlying worst-case approximation factor than prior schemes.