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

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On Lattices, Learning with Errors, Random Linear Codes, and Cryptography (2005)

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by Oded Regev

Venue:	In STOC
Citations:	361 - 6 self

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@INPROCEEDINGS{Regev05onlattices,,
    author = {Oded Regev},
    title = {On Lattices, Learning with Errors, Random Linear Codes, and Cryptography},
    booktitle = {In STOC},
    year = {2005},
    pages = {84--93},
    publisher = {ACM Press}
}

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Abstract

Our main result is a reduction from worst-case lattice problems 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 quantum. 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 classical. 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 SVP and SIVP. Previous lattice-based public-key cryptosystems such as the one by Ajtai and Dwork were based only on unique-SVP, a special case of SVP. The new cryptosystem is much more efficient than previous cryptosystems: the public key is of size Õ(n2) and encrypting a message increases its size by a factor of Õ(n) (in previous cryptosystems 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). 1

Keyphrases

random linear code learning problem main result public key previous cryptosystems previous lattice-based public-key cryptosystems strong indication random bit string worst-case lattice problem party share main open question quantum algorithm new cryptosystem efficient solution natural extension worst-case quantum hardness public-key cryptosystem special case certain learning problem error problem

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