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**On distribution functions of sequences generated by scalar and mixed product.**
*(English)*
Zbl 1061.11042

In the first part of the paper the author is interested in the quantity \(g_s(t)\), the Lebesgue measure of the set \(\{(\mathbf b,x)\in [0,1]^{2s}: \mathbf b \cdot x < t\}\), where \(s\) is an integer, \(s\geq 1\), \(\mathbf b\) and \(x\) are \(s\)-dimensional vectors with components in \([0,1]\), \(\mathbf b \cdot x\) is their scalar product and \(t\) is a real number, \(0\leq t\leq s\). It is known that
\[
g_1(t)=t - t \log, t, \quad t\in [0,1].
\]
The author calculates \(g_2(t)\) and \(g_3(t)\) for \(0\leq t \leq 1\). To shed some light on the rather complicated formulas, he mentions an unpublished communication by Laurent Habsieger (Lyon 1) relating \(g_s\) to values of the Riemann \(\zeta \) function on positive integers. Nothing is known when \(1\leq t\leq s\).

The author is interested in the \(g_s\) functions for theoretical and practical reasons. If \(\mathbb B = (\mathbf b_n)_{n\geq 1}\) and \(\mathbb X =(x_n)_{n\geq 1}\) are sequences of integers, the asymptotic distribution function of the scalar product is defined by \[ g(t)=\lim _{N\to \infty } \frac {\# \{n\leq N: \mathbf b_n \cdot x_n <t\}} N. \] If the sequences \(\mathbb B\) and \(\mathbb X\) are uniformly distributed in \([0,1]^s\) and statistically independent, then \(g(t)\) is equal to \(g_s(t)\).

In the second part of the paper the author proves analogous results for the distribution function of the sequence of absolute values \(| \bigl (\mathbf b^{(1)}_n \times \mathbf b ^{(2)}_n\bigr ) \cdot x_n| \) of mixed products where again \(\mathbb B^{(1)}, \mathbb B^{(2)}\) and \(X\) are uniformly distributed and statistically independent sequences in \([0,1]^s\), \(s=2\) or 3.

Applications to cryptosystems are presented in the last two sections.

The author is interested in the \(g_s\) functions for theoretical and practical reasons. If \(\mathbb B = (\mathbf b_n)_{n\geq 1}\) and \(\mathbb X =(x_n)_{n\geq 1}\) are sequences of integers, the asymptotic distribution function of the scalar product is defined by \[ g(t)=\lim _{N\to \infty } \frac {\# \{n\leq N: \mathbf b_n \cdot x_n <t\}} N. \] If the sequences \(\mathbb B\) and \(\mathbb X\) are uniformly distributed in \([0,1]^s\) and statistically independent, then \(g(t)\) is equal to \(g_s(t)\).

In the second part of the paper the author proves analogous results for the distribution function of the sequence of absolute values \(| \bigl (\mathbf b^{(1)}_n \times \mathbf b ^{(2)}_n\bigr ) \cdot x_n| \) of mixed products where again \(\mathbb B^{(1)}, \mathbb B^{(2)}\) and \(X\) are uniformly distributed and statistically independent sequences in \([0,1]^s\), \(s=2\) or 3.

Applications to cryptosystems are presented in the last two sections.

Reviewer: Georges Grekos (Saint Étienne)

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### References:

[1] | DRMOTA M.-TICHY R. F.: Sequences, Discrepancies and Applications. Lecture Notes in Math. 1651, Springer-Verlag, Berlin-Heidelberg, 1997. · Zbl 0877.11043 · doi:10.1007/BFb0093404 |

[2] | KENDALL D. G.-MORAN P. A. P.: Geometrical Probabilities. (Russian translation), Izd. Nauka, Moscow, 1972. |

[3] | MENEZES A.-OORSCHOT P. van-VANSTONE S.: Handbook of Applied Cryptography. CRC Press (electronic version), 1996. · Zbl 0868.94001 |

[4] | RYSHIK I. M.-GRADSTEIN I. S.: Tables of Series, Products, and Integrals. ( · Zbl 0080.33703 |

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