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

Research output: Contribution to journalArticle

Abstract

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., nonquantum). 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 (n2) and encrypting a message increases its size by a factor of (n) (in previous cryptosystems these values are (n4) and (n 2), respectively). In fact, under the assumption that all parties share a random bit string of length (n2), the size of the public key can be reduced to (n).

Original languageEnglish (US)
Article number1568324
JournalJournal of the ACM
Volume56
Issue number6
DOIs
StatePublished - Sep 1 2009

Keywords

  • Average-case hardness
  • Cryptography
  • Lattice
  • Public key encryption
  • Quantum computation

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence

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