Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika
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Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2009, Number 4, Pages 3–7 (Mi vmumm882)  

This article is cited in 5 scientific papers (total in 5 papers)

Mathematics

The complexity and depth of Boolean circuits for multiplication and inversion in some fields $GF(2^n)$

S. B. Gashkova, I. S. Sergeevb

a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Research Institute "Kvant", Moscow

Abstract: Let $n=(p-1)\cdot p^k,$ where $p$ is a prime number such that $2$ is a primitive root modulo $p$, $2^{p-1}-1$ is not divided by $p^2$. For a standard basis of the field $GF(2^n)$, a multiplier of complexity $ O(\log \log p)n\log n\log\log_p n$ and an invertor of complexity $ O(\log p\log\log p)n\log n\log\log_p n$ are constructed. In particular, in the case $p=3$ the upper bound
$$ 5\frac{5}{8}n\log_3 n\log_2\log_3 n+O(n\log n) $$
for the multiplication complexity and the asymptotically $2,5$ times greater bound for the inversion complexity are obtained.

Key words: Boolean schemes, finite fields, multiplier, invertor.

Funding Agency Grant Number
Russian Foundation for Basic Research 08-01-00863
08-01-00632-ą
Russian Academy of Sciences - Federal Agency for Scientific Organizations


Full text: PDF file (213 kB)

Bibliographic databases:
UDC: 519.95
Received: 24.12.2008

Citation: S. B. Gashkov, I. S. Sergeev, “The complexity and depth of Boolean circuits for multiplication and inversion in some fields $GF(2^n)$”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2009, no. 4, 3–7

Citation in format AMSBIB
\Bibitem{GasSer09}
\by S.~B.~Gashkov, I.~S.~Sergeev
\paper The complexity and depth of Boolean circuits for multiplication and inversion in some fields $GF(2^n)$
\jour Vestnik Moskov. Univ. Ser.~1. Mat. Mekh.
\yr 2009
\issue 4
\pages 3--7
\mathnet{http://mi.mathnet.ru/vmumm882}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2657270}
\zmath{https://zbmath.org/?q=an:1304.94131}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. S. Sergeev, “Regulyarnye otsenki slozhnosti umnozheniya mnogochlenov i usechennogo DPF”, PDM, 2011, no. 4(14), 72–88  mathnet
    2. S. B. Gashkov, I. S. Sergeev, “Complexity of computation in finite fields”, J. Math. Sci., 191:5 (2013), 661–685  mathnet  crossref
    3. S. B. Gashkov, I. S. Sergeev, “On complexity and depth of Boolean circuits for multiplication and inversion over finite fields of characteristic 2”, Discrete Math. Appl., 23:1 (2013), 1–37  mathnet  crossref  crossref  mathscinet  elib  elib
    4. S. B. Gashkov, I. B. Gashkov, “Fast algorithm of square rooting in some odd charactećistic finite field”, Moscow University Mathematics Bulletin, Moscow University Måchanics Bulletin, 73:5 (2018), 176–181  mathnet  crossref  mathscinet  zmath  isi
    5. S. B. Gashkov, “Bystrye algoritmy resheniya uravnenii stepeni ne vyshe chetvertoi v nekotorykh konechnykh polyakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2021, no. 3, 22–31  mathnet
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