Section: New Results
Cryptography and lattices
Group signatures
Group signatures are cryptographic primitives where users can anonymously sign messages in the name of a population they belong to. Gordon et al. (Asiacrypt 2010) suggested the first realization of group signatures based on lattice assumptions in the random oracle model. A significant drawback of their scheme is its linear signature size in the cardinality N of the group. A recent extension proposed by Camenisch et al. (SCN 2012) suffers from the same overhead.
F. Laguillaumie, A. Langlois, B. Libert (Technicolor), and D. Stehlé described in [24] the first lattice-based group signature schemes where the signature and public key sizes are essentially logarithmic in N (for any fixed security level). Their basic construction only satisfies a relaxed definition of anonymity (just like the Gordon et al. system) but readily extends into a fully anonymous group signature (i.e., that resists adversaries equipped with a signature opening oracle). They proved the security of their schemes in the random oracle model under the SIS and LWE assumptions.
Classical hardness of learning with errors
Z. Brakerski (Stanford U.), A. Langlois, C. Peikert (Georgia Institute of Technology), O. Regev (Courant Institute, New York U.), and D. Stehlé showed in [16] that the Learning with Errors (LWE) problem is classically at least as hard as standard worst-case lattice problems, even with polynomial modulus. Previously this was only known under quantum reductions. Their techniques capture the tradeoff between the dimension and the modulus of LWE instances, leading to a much better understanding of the landscape of the problem. The proof is inspired by techniques from several recent cryptographic constructions, most notably fully homomorphic encryption schemes.
Improved Zero-knowledge Proofs of Knowledge for the ISIS Problem, and Applications
In all existing efficient proofs of knowledge of a solution to the
infinity norm Inhomogeneous Small Integer Solution ISIS
Decoding by Embedding: Correct Decoding Radius and DMT Optimality
In lattice-coded multiple-input multiple-output (MIMO) systems,
optimal decoding amounts to solving the closest vector problem
(CVP). Embedding is a powerful technique for the approximate CVP, yet
its remarkable performance is not well understood. In [8] ,
C. Ling (Imperial College, London), L. Luzzi (ENSEA, U. Cergy Pontoise), and D. Stehlé
analyzed the embedding technique from a bounded distance decoding (BDD)
viewpoint. They proved that the Lenstra, Lenstra and Lovász (LLL)
algorithm can achieve
A New View on HJLS and PSLQ: Sums and Projections of Lattices
The HJLS and PSLQ algorithms are the de facto standards for discovering non-trivial integer relations between a given tuple of real numbers. In [19] , J. Chen, D. Stehlé, and G. Villard provided a new interpretation of these algorithms, in a more general and powerful algebraic setup: they view them as special cases of algorithms that compute the intersection between a lattice and a vector subspace. Further, they extracted from them the first algorithm for manipulating finitely generated additive subgroups of a Euclidean space, including projections of lattices and finite sums of lattices. They adapted the analyses of HJLS and PSLQ to derive correctness and convergence guarantees. They also investigated another approach based on embedding the input in a higher dimensional lattice and calling the LLL lattice reduction algorithm.