Linearly Homomorphic Signatures From Lattices

Linearly homomorphic signatures (LHSs) allow any entity to linearly combine a set of signatures and to provide authentication service for the corresponding (combined) data. The public key of the current known LHSs from lattices in the standard model requires O(l) matrices and O(k) vectors, where l is the length of file identifier and k is the maximum data set size that linear functions support. In this paper, we construct two lattice-based LHS schemes with provable security in the standard model and both schemes can authenticate vectors defined over finite field. First, we present a basic LHS scheme satisfying selective security, based on the full-rank difference hash functions. Second, we modify the chameleon hash function constructed by (Cash, D., Hofheinz, D., Kiltz, E. and Peikert, C. (2010) Bonsai Trees, or How to Delegate a Lattice Basis. In Proc. EUROCRYPT 10, Monaco/French Riviera, May 30 to June 3, pp. 523-552. Springer, Berlin) to construct a linearly homomorphic chameleon hash function (LHCHF), which can be applied to all transformations from selectively secure LHS scheme that authenticates vectors defined over finite field F-p (p = poly(n)) to fully secure one, except for a new one that authenticates vectors defined over a small field. Starting from LHCFH and the basic scheme as above, we obtain a fully secure LHS scheme. Both schemes can be used to sign multiple files and have relatively short public keys consisting of O(1) matrices and O(k) vectors.