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

Our main result is a reduction from worst-case lattice problems such as GAPSVP 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 quantum. Hence, an efficient solution to the learning problem implies a quantum algorithm for GAPSVP and SIVP. A main open question is whether this reduction can be made classical (i.e., non-quantum). We also present a (classical) public-key cryptosystem whose security is based on the hardness of the learning problem. By the main result, its security is also based on the worst-case quantum hardness of GAPSVP and SIVP. The new cryptosystem is much more efficient than previous lattice-based cryptosystems: the public key is of size ~O(n2 ) and encrypting a message increases its size...

• Type: Conference
• Year: 2005
• Where: STOC
• Authors: Oded Regev
• Keywords: Algorithms, Learning Problem, Public Key, Random Bit String, STOC 2005