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Algorithms for the approximate common divisor problem

Part of: Computational number theory Communication, information

Published online by Cambridge University Press:  26 August 2016

Steven D. Galbraith ,
Shishay W. Gebregiyorgis  and
Sean Murphy
Steven D. Galbraith
Affiliation:
Mathematics Department, University of Auckland, New Zealand email [email protected]
Shishay W. Gebregiyorgis
Affiliation:
Mathematics Department, University of Auckland, New Zealand email [email protected]
Sean Murphy
Affiliation:
Royal Holloway, University of London, United Kingdom email [email protected]

Abstract

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The security of several homomorphic encryption schemes depends on the hardness of variants of the approximate common divisor (ACD) problem. We survey and compare a number of lattice-based algorithms for the ACD problem, with particular attention to some very recently proposed variants of the ACD problem. One of our main goals is to compare the multivariate polynomial approach with other methods. We find that the multivariate polynomial approach is not better than the orthogonal lattice algorithm for practical cryptanalysis.

We also briefly discuss a sample-amplification technique for ACD samples and a pre-processing algorithm similar to the Blum–Kalai–Wasserman algorithm for learning parity with noise. The details of this work are given in the full version of the paper.

MSC classification

Primary: 11Y16: Algorithms; complexity
Secondary: 94A60: Cryptography
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 58 - 72
Copyright
© The Author(s) 2016 

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