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STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
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CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
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ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
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We devise an algorithm, L1, with the following specifications: It takes as input an arbitrary basis B=(bi)i ∈ Zd x d of a Euclidean lattice L; It computes a basis of L which is reduced for a mild modification of the Lenstra-Lenstra-Lovász reduction; It terminates in time O(d5+ε β + dω+1+ε β1+ε) where β = log max |bi| (for any ε0 and ω is a valid exponent for matrix multiplication). This is the first LLL-reducing algorithm with a time complexity that is quasi-linear in β and polynomial in d. The backbone structure of L1 is able to mimic the Knuth-Schönhage fast gcd algorithm thanks to a combination of cutting-edge ingredients. First the bit-size of our lattice bases can be decreased via truncations whose validity are backed by recent numerical stability results on the QR matrix factorization. Also we establish a new framework for analyzing unimodular transformation matrices which reduce shifts of reduced bases, this includes bit-size control and new perturbation tools. We illustrate the power of this framework by generating a family of reduction algorithms.