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**2007/432**

**Title:**

Trapdoors for Hard Lattices and New Cryptographic Constructions

**Authors:**
Craig Gentry, Chris Peikert, Vinod Vaikuntanathan

**Abstract:**

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 \emph{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 \emph{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.

**ePrint:**
https://eprint.iacr.org/2007/432

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