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Diskr. Mat., 1998, Volume 10, Issue 4, Pages 35–38 (Mi dm442)  

This article is cited in 1 scientific paper (total in 1 paper)

Simplified justification of the probabilistic Miller–Rabin test for primality

S. B. Gashkov


Abstract: Let $m$ be a positive integer and $\mathbb Z_m^*$ be the set of all positive integeres which are no greater than $m$ and relatively prime to $m$. A number $s\in\mathbb Z_m^*$ is called a witness of primality of $m$ if the sequence
$$ s^{(m-1)2^{-i}} \pmod m,\quad i = 0,1,\ldots, r,\quad m-1 = 2^{r}t, $$
where $t$ is odd, consists only of ones, or begins with ones and continues by minus one and, may be, then by other integers.
We give a simple proof of the following known assertion that is a ground of the Miller–Rabin primality test: The cardinality of the set of witnesses of primality of a composite $m$ is no greater than $\varphi(m)/4$, where $\varphi(m)$ is Euler's totient function.
This work was supported by the Russian Foundation for Basic Research, grant 96–01–068.

DOI: https://doi.org/10.4213/dm442

Full text: PDF file (359 kB)

English version:
Discrete Mathematics and Applications, 1998, 8:6, 545–548

Bibliographic databases:

UDC: 519.7
Received: 02.02.1998

Citation: S. B. Gashkov, “Simplified justification of the probabilistic Miller–Rabin test for primality”, Diskr. Mat., 10:4 (1998), 35–38; Discrete Math. Appl., 8:6 (1998), 545–548

Citation in format AMSBIB
\Bibitem{Gas98}
\by S.~B.~Gashkov
\paper Simplified justification of the probabilistic Miller--Rabin test for primality
\jour Diskr. Mat.
\yr 1998
\vol 10
\issue 4
\pages 35--38
\mathnet{http://mi.mathnet.ru/dm442}
\crossref{https://doi.org/10.4213/dm442}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1673119}
\zmath{https://zbmath.org/?q=an:0985.11065}
\transl
\jour Discrete Math. Appl.
\yr 1998
\vol 8
\issue 6
\pages 545--548


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Vasilenko O.N., “On the Miller-Rabin algorithm”, Vestnik Moskovskogo Universiteta Seriya 1 Matematika Mekhanika, 2000, no. 2, 41–42  mathscinet  zmath  isi
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