We develop secure \\emph{threshold} protocols for two importantoperations in lattice cryptography, namely, generating a hard lattice

$\\Lambda$ together with a ``strong\'\' trapdoor, and sampling from a

discrete Gaussian distribution over a desired coset of $\\Lambda$ using

the trapdoor. These are the central operations of many cryptographic

schemes: for example, they are exactly the key-generation and signing

operations (respectively) for the GPV signature scheme, and they are

the public parameter generation and private key extraction operations

(respectively) for the GPV IBE. We also provide a protocol for

trapdoor delegation, which is used in lattice-based hierarchical IBE

schemes. Our work therefore directly transfers all these systems to

the threshold setting.

Our protocols provide information-theoretic (i.e., statistical)

security against adaptive corruptions in the UC framework, and they

are private and robust against an

optimal number of semi-honest or malicious parties. Our Gaussian

sampling protocol is both noninteractive and efficient, assuming

either a trusted setup phase (e.g., performed as part of key

generation) or a sufficient amount of interactive but offline

precomputation, which can be performed before the inputs to the

sampling phase are known.