Short Signatures with Short Public Keys from Homomorphic Trapdoor Functions

We present a lattice-based stateless signature scheme provably secure in the standard model. Our scheme has a constant number of matrices in the public key and a single lattice vector (plus a tag) in the signatures. The best previous lattice-based encryption schemes were the scheme of Ducas and Micciancio (CRYPTO 2014), which required a logarithmic number of matrices in the public key and that of Bohl et. al (J. of Cryptology 2014), which required a logarithmic number of lattice vectors in the signature. Our main technique involves using fully homomorphic computation to compute a degree \(d\) polynomial over the tags hidden in the matrices in the public key. In the scheme of Ducas and Micciancio, only functions linear over the tags in the public key matrices were used, which necessitated having \(d\) matrices in the public key.

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