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Fast lattice reduction for $ \mathbf{F}_2$-linear pseudorandom number generators


Authors: Shin Harase, Makoto Matsumoto and Mutsuo Saito
Journal: Math. Comp. 80 (2011), 395-407
MSC (2010): Primary 11K45; Secondary 65C10
DOI: https://doi.org/10.1090/S0025-5718-2010-02391-9
Published electronically: June 18, 2010
MathSciNet review: 2728986
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Abstract: Sequences generated by an $ {\bf F}_2$-linear recursion have wide applications, in particular, pseudorandom number generation. The dimension of equidistribution with $ v$-bit accuracy is a most important criterion for the uniformity of the generated sequence. The fastest known method for computing these dimensions is proposed by Couture and L'Ecuyer, based on Lenstra's lattice basis reduction and the dual lattice to the lattice of vector-valued generating functions (with components in the formal power series $ {\bf F}_2[[t^{-1}]]$) associated to the output $ \mathbf{F}_2$-vector sequence. In this paper we propose a similar but faster algorithm, where (1) the state space is used to represent vectors with components in the formal power series, (2) the dual lattice is not necessary, and (3) Lenstra reduction is replaced with a simpler basis reduction. The computational complexity of our method is smaller than for the Couture-L'Ecuyer method. Experiments show that our method improves the speed by a factor of 10 for Mersenne Twister MT19937 and for WELL generators with state sizes of 19937 bits and 44497 bits.


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Additional Information

Shin Harase
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan
Email: sharase@orange.ocn.ne.jp

Makoto Matsumoto
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan
Email: matumoto@ms.u-tokyo.ac.jp

Mutsuo Saito
Affiliation: Department of Mathematics, Hiroshima University, Hiroshima, Japan
Email: saito@math.sci.hiroshima-u.ac.jp

DOI: https://doi.org/10.1090/S0025-5718-2010-02391-9
Received by editor(s): July 15, 2009
Received by editor(s) in revised form: October 18, 2009
Published electronically: June 18, 2010
Additional Notes: The first author was partially supported by Grant-in-Aid for JSPS Fellows 21$·$4427
The second author was partially supported by JSPS Grant-in-Aid for Challenging Exploratory Research 21654017, Scientific Research (A) 19204002 and JSPS Core-to-Core Program 18005.
The third author was partially supported by JSPS Grant-in-Aid for Challenging Exploratory Research 21654004
Article copyright: © Copyright 2010 American Mathematical Society

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