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- W3048502882 abstract "We describe two very efficient polynomial-time algorithms for reducing module lattices defined over arbitrary cyclotomic fields that solve the $$gamma $$ -Hermite Module-SVP problem. They both exploit the structure of tower fields and the second one also uses the symplectic geometry existing in these fields. We conjecture that a rank-2 module over a cyclotomic field of degree n with B-bit coefficients can be heuristically reduced within approximation factor $$2^{widetilde{text {O}}left( nright) }$$ in time $$widetilde{text {O}}left( n^2Bright) $$ . In the symplectic algorithm, if the condition number C of the input matrix is large enough, this complexity shrinks to $$widetilde{text {O}}left( n^{log _2 3}Cright) $$ . In cryptography, matrices are well-conditioned and we can take $$C=B$$ , but in the worst case, C can be as large as nB. This last result is particularly striking as for some matrices, we can go below the $$n^2B$$ swaps lower bound given by the analysis of LLL based on the potential. These algorithms are parallel and we provide a full implementation. We apply them on multilinear cryptographic concrete parameters by reducing matrices of dimension 4096 with 6675-bit integers in 4 days. Finally, we give a quasicubic time for the Gentry-Szydlo algorithm and run it in dimension 1024. It requires efficient ideal multiplications which need fast lattice reductions." @default.
- W3048502882 created "2020-08-18" @default.
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- W3048502882 date "2020-01-01" @default.
- W3048502882 modified "2023-09-24" @default.
- W3048502882 title "Fast Reduction of Algebraic Lattices over Cyclotomic Fields" @default.
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- W3048502882 doi "https://doi.org/10.1007/978-3-030-56880-1_6" @default.
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