On lattices, learning with errors, random linear codes, and cryptography

Our main result is a reduction from worst-case lattice prob-lems such as SVP and SIVP to a certain learning problem. This learning problem is a natural extension of the ‘learning from parity with error ’ problem to higher moduli. It can also be viewed as the problem of decoding from a random linear code. This, we believe, gives a strong indication that these problems are hard. Our reduction, however, is quan-tum. Hence, an efficient solution to the learning problem implies a quantum algorithm for SVP and SIVP. A main open question is whether this reduction can be made classi-cal. Using the main result, we obtain a public-key cryptosys-tem whose hardness is based on the worst-case quantum hardness of SVP and SIVP. Previous lattice-based public-key cryptosystems such as the one by Ajtai and Dwork were only based on unique-SVP, a special case of SVP. The new cryptosystem is much more efficient than previous cryp-tosystems: the public key is of size Õ(n2) and encrypting a message increases its size by Õ(n) (in previous cryptosys-tems these values are Õ(n4) and Õ(n2), respectively). In fact, under the assumption that all parties share a random bit string of length Õ(n2), the size of the public key can be reduced to Õ(n)