Abstract
The GGH Graded Encoding Scheme[9], based on ideal lattices, is the first plausible approximation to a cryptographic multilinear map. Unfortunately, using the security analysis in[9], 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[9] 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 Ω(n logn) 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(λ 5 logλ) for the GGH scheme to O(λlog2 λ) in GGHLite, with respect to the security parameter λ (for a constant multilinearity parameter κ).
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Langlois, A., Stehlé, D., Steinfeld, R. (2014). GGHLite: More Efficient Multilinear Maps from Ideal Lattices. In: Nguyen, P.Q., Oswald, E. (eds) Advances in Cryptology – EUROCRYPT 2014. EUROCRYPT 2014. Lecture Notes in Computer Science, vol 8441. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55220-5_14
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