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Published in:
2012 | OriginalPaper | Chapter
Commitments and Efficient Zero-Knowledge Proofs from Learning Parity with Noise
Authors : Abhishek Jain, Stephan Krenn, Krzysztof Pietrzak, Aris Tentes
Published in: Advances in Cryptology – ASIACRYPT 2012
Publisher: Springer Berlin Heidelberg
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We construct a perfectly binding string commitment scheme whose security is based on the learning parity with noise (
LPN
) assumption, or equivalently, the hardness of decoding random linear codes. Our scheme not only allows for a simple and efficient zero-knowledge proof of knowledge for committed values (essentially a Σ-protocol), but also for such proofs showing any kind of relation amongst committed values, i.e., proving that messages
m
0
,…,
m
u
, are such that
m
0
=
C
(
m
1
,…,
m
u
) for any circuit
C
.
To get soundness which is exponentially small in a security parameter
t
, and when the zero-knowledge property relies on the
LPN
problem with secrets of length ℓ, our 3 round protocol has communication complexity
${\mathcal O}(t|C|\ell\log(\ell))$
and computational complexity of
${\mathcal O}(t|C|\ell)$
bit operations. The hidden constants are small, and the computation consists mostly of computing inner products of bit-vectors.