# GGHLite: More Efficient Multilinear Maps from Ideal Lattices

### Adeline Langlois, Damien Stehlé and Ron Steinfeld

Abstract: The GGH Graded Encoding Scheme \cite{GGH13}, based on ideal lattices, is the first plausible approximation to a cryptographic multilinear map. Unfortunately, using the security analysis in \cite{GGH13}, the scheme requires very large parameters to provide security for its underlying encoding re-randomization process. Our main contributions are to formalize, simplify and improve the efficiency and the security analysis of the re-randomization process in the GGH construction. This results in a new construction that we call GGHLite. In particular, we first lower the size of a standard deviation parameter of the re-randomization process of \cite{GGH13} from exponential to polynomial in the security parameter. This first improvement is obtained via a finer security analysis of the drowning step of re-randomization, in which we apply the Rényi divergence instead of the conventional statistical distance as a measure of distance between distributions. Our second improvement is to reduce the number of randomizers needed from Omega(n log n) to 2, where n is the dimension of the underlying ideal lattices. These two contributions allow us to decrease the bit size of the public parameters from O(lambda^5 log lambda) for the GGH scheme to O(lambda log^2 lambda) in GGHLite, with respect to the security parameter lambda (for a constant multilinearity parameter kappa).