## 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 To shed some light on the rather complicated formulas, he mentions an unpublished communication by Laurent Habsieger (Lyon 1) relating $$g_s$$ to values of the Riemann $$\zeta$$ function on positive integers.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.

### MSC:

 11K31 Special sequences 94A60 Cryptography

### Keywords:

distribution function; cryptosystems
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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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