Trapdoors For Hard Lattices And New Cryptographic Constructions

We show how to construct a variety of "trapdoor" cryptographic tools assuming the worst-case hardness of Standard lattice problems (such as approximating the length of the shortest nonzero vector to within certain polynomial factors). Our contributions include a new notion of trapdoor function with preimage sampling, simple and efficient; "hash-and-sign" digital signature schemes, and identity-based encryption.A core technical component, of our constructions is all efficient algorithm that, given a basis of an arbitrary lattice, samples lattice points from a discrete Caussian probability distribution whose standard deviation is essentially the length of the longest Gram-Schmidt vector of the basis. A crucial security property is that the output distribution of the algorithm is oblivious to the particular geometry of the given basis.