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Our main result is a reduction from worst-case lattice problems such asGapSVP and SIVP to a certain learning problem. This learning problem is anatural extension of the `learning from parity with error' problem to highermoduli. 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, is quantum. Hence, an efficient solution to thelearning problem implies a quantum algorithm for GapSVP and SIVP. A main openquestion is whether this reduction can be made classical (i.e., non-quantum).  We also present a (classical) public-key cryptosystem whose security is basedon the hardness of the learning problem. By the main result, its security isalso based on the worst-case quantum hardness of GapSVP and SIVP. The newcryptosystem is much more efficient than previous lattice-based cryptosystems:the public key is of size $\tilde{O}(n^2)$ and encrypting a message increasesits size by a factor of $\tilde{O}(n)$ (in previous cryptosystems these valuesare $\tilde{O}(n^4)$ and $\tilde{O}(n^2)$, respectively). In fact, under theassumption that all parties share a random bit string of length$\tilde{O}(n^2)$, the size of the public key can be reduced to $\tilde{O}(n)$.

On Lattices, Learning with Errors, Random Linear Codes, and Cryptography