![](bg6.png)
The tests chosen are powerful and operate well on
large sample sizes, with each analysing different as-
pects within the discrete Gaussian distribution. Fail-
ure in any of these tests indicates a deviation from
the target distribution, which is therefore evidence of
an incorrectly performing discrete Gaussian sampler.
The software for GLITCH is made available online
(https://github.com/jameshoweee/glitch), which also
provides sample data for discrete Gaussian samplers;
which are able to be tested upon.
The full version of this paper is available at (Howe
and O’Neill, 2017), which includes more concise de-
tails on the statistical formulae used as well as exam-
ple results discussions.
REFERENCES
Ajtai, M. (1996). Generating hard instances of lattice prob-
lems (extended abstract). In STOC, pages 99–108.
Anderson, T. W. and Darling, D. A. (1952). Asymptotic
theory of certain ”goodness of fit” criteria based on
stochastic processes. The Annals of Mathematical
Statistics, 23(2):193–212.
Anderson, T. W. and Darling, D. A. (1954). A test of good-
ness of fit. Journal of the American Statistical Associ-
ation, 49(268):765–769.
Balanda, K. P. and MacGillivray, H. (1988). Kurtosis: a
critical review. The American Statistician, 42(2):111–
119.
Bassham III, L. E., Rukhin, A. L., Soto, J., Nechvatal, J. R.,
Smid, M. E., Barker, E. B., Leigh, S. D., Levenson,
M., Vangel, M., Banks, D. L., Heckert, N. A., Dray,
J. F., and Vo, S. (2010). SP 800-22 Rev. 1a. A Statisti-
cal Test Suite for Random and Pseudorandom Number
Generators for Cryptographic Applications. Technical
report, Gaithersburg, MD, United States.
Campagna, M., Chen, L., Dagdelen,
¨
O., Ding, J., Fernick,
J., Gisin, N., Hayford, D., Jennewein, T., L
¨
utkenhaus,
N., Mosca, M., et al. (2015). Quantum safe cryptog-
raphy and security. ETSI White Paper, (8).
CESG (2016). Quantum key distribution: A CESG white
paper.
CNSS (2015). Use of public standards for the secure shar-
ing of information among national security systems.
Committee on National Security Systems: CNSS Ad-
visory Memorandum, Information Assurance 02-15.
D’Agostino, R. B., Belanger, A., and D’Agostino Jr, R. B.
(1990). A suggestion for using powerful and infor-
mative tests of normality. The American Statistician,
44(4):316–321.
Ducas, L., Durmus, A., Lepoint, T., and Lyubashevsky, V.
(2013). Lattice signatures and bimodal Gaussians. In
CRYPTO (1), pages 40–56. Full version: https://
eprint.iacr.org/2013/383.pdf.
Dwarakanath, N. C. and Galbraith, S. D. (2014). Sampling
from discrete Gaussians for lattice-based cryptogra-
phy on a constrained device. Appl. Algebra Eng. Com-
mun. Comput., pages 159–180.
Field, A. (2009). Discovering statistics using SPSS. Sage
publications.
Gentry, C., Peikert, C., and Vaikuntanathan, V. (2008).
Trapdoors for hard lattices and new cryptographic
constructions. In STOC, pages 197–206.
Hinton, P. R. (2014). Statistics explained. Routledge.
Howe, J., Khalid, A., Rafferty, C., Regazzoni, F., and
O’Neill, M. (2016). On Practical Discrete Gaus-
sian Samplers For Lattice-Based Cryptography. IEEE
Transactions on Computers.
Howe, J. and O’Neill, M. (2017). GLITCH: A Discrete
Gaussian Testing Suite For Lattice-Based Cryptogra-
phy. Cryptology ePrint Archive, Report 2017/438.
http://eprint.iacr.org/2017/438.
Howe, J., P
¨
oppelmann, T., O’Neill, M., O’Sullivan, E.,
and G
¨
uneysu, T. (2015). Practical lattice-based digital
signature schemes. ACM Transactions on Embedded
Computing Systems, 14(3):24.
Jarque, C. M. and Bera, A. K. (1987). A test for normality
of observations and regression residuals. International
Statistical Review, pages 163–172.
Joanes, D. N. and Gill, C. A. (1998). Comparing measures
of sample skewness and kurtosis. Journal of the Royal
Statistical Society: Series D (The Statistician), 47(1).
Kim, H.-Y. (2013). Statistical notes for clinical re-
searchers: assessing normal distribution (2) using
skewness and kurtosis. Restorative dentistry & en-
dodontics, 38(1):52–54.
Kolmogorov, A. N. (1956). Foundations of the theory of
probability (2nd ed.). Chelsea Publishing Co., New
York.
Lindner, R. and Peikert, C. (2011). Better key sizes (and
attacks) for LWE-based encryption. In CT-RSA, pages
319–339.
Lyubashevsky, V., Peikert, C., and Regev, O. (2013). On
ideal lattices and learning with errors over rings. J.
ACM, 60(6):43.
Marsaglia, G. (1985). A current view of random number
generators. In Computer Science and Statistics, Six-
teenth Symposium on the Interface. Elsevier Science
Publishers, North-Holland, Amsterdam, pages 3–10.
Marsaglia, G. (1993). A current view of random numbers.
In Billard, L., editor, Computer Science and Statistics:
Proceedings of the 16th Symposium on the Interface,
volume 36, pages 105–110. Elsevier Science Publish-
ers B. V.
Marsaglia, G. (1996). DIEHARD: A battery of tests
of randomness. http://www.stat.fsu.edu/pub/
diehard/.
Moody, D. (2016). Post-quantum cryptography: NIST’s
plan for the future. Talk given at PQCrypto ’16 Con-
ference, 23-26 February 2016, Fukuoka, Japan.
Peikert, C. (2010). An efficient and parallel Gaussian sam-
pler for lattices. In CRYPTO, pages 80–97.
P
¨
oppelmann, T. and G
¨
uneysu, T. (2014). Area optimiza-
tion of lightweight lattice-based encryption on recon-
figurable hardware. In ISCAS, pages 2796–2799.
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