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 preimage sampleable functions, simple and efficient “hash-and-sign” digital signature schemes, and identity-based encryption. A core technical component of our constructions is an efficient algorithm that, given a basis of an arbitrary lattice, samples lattice points from a discrete Gaussian 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. ∗ Supported by the Herbert Kunzel Stanford Graduate Fellowship. †

• Type: Journal
• Year: 2007
• Where: ECCC
• Authors: Craig Gentry, Chris Peikert, Vinod Vaikuntanathan
• Keywords: Certain Polynomial Factors, ECCC 2007, Shortest Nonzero Vector, Standard Lattice Problems