Towards Tightly Secure Lattice Short Signature and Id-Based Encryption.

Constructing short signatures with tight security from standard assumptions is a long-standing open problem. We present an adaptively secure, short and stateless signature scheme, featuring a constant security loss relative to a conservative hardness assumption, Short Integer Solution SIS, and the security of a concretely instantiated pseudorandom function PRF. This gives a class of tightly secure short lattice signature schemes whose security is based on SIS and the underlying assumption of the instantiated PRF. Our signature construction further extends to give a class of tightly and adaptively secure \"compact\" Identity-Based Encryption IBE schemes, reducible with constant security loss from Regev's vanilla Learning With Errors LWE hardness assumption and the security of a concretely instantiated PRF. Our approach is a novel combination of a number of techniques, including Katz and Wang signature, Agrawal et al. lattice-based secure IBE, and Boneh et al. key-homomorphic encryption. Our results, at the first time, eliminate the dependency between the number of adversary's queries and the security of short signature/IBE schemes in the context of lattice-based cryptography. They also indicate that tightly secure PRFs with constant security loss would imply tightly, adaptively secure short signature and IBE schemes with constant security loss.