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On Lattices, Learning with Errors, Random Linear Codes ...

On Lattices, Learning with Errors, Random Linear Codes , and CryptographyOded Regev May 2, 2009 AbstractOur main result is a reduction from worst-case lattice problems such asGAPSVPandSIVPto acertain Learning problem. This Learning problem is a natural extension of the Learning from parity witherror problem to higher moduli. It can also be viewed as the problem of decoding from a Random linearcode. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, isquantum. Hence, an efficient solution to the Learning problem implies aquantumalgorithm forGAPSVPandSIVP.

best known polynomial time algorithms for them yield only mildly subexponential approximation factors [24, 38, 5]. It is conjectured that there is no classical (i.e., non-quantum) polynomial time algorithm that approximates them to within any polynomial factor. Lattice-based constructions of one-way functions, such

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