The authors analyze a family of finite binary sequences $E_N$ of length $N$. Two measures of pseudorandomness have been introduced by {\it C. Mauduit} and {\it A. Sárközy} in [Acta Arith. 82, 365‒377 (1997; Zbl 0886.11048)]: the well-distribution measure $W(E_N)$ and the correlation measure of order $k$, $C_k(E_N)$. A construction of sequences $E_N$ has been proposed in the same article. It depends on a parameter $p$ (an odd prime) and uses the Legendre symbol. An extended construction (called $Q$-sequences) has been given by Goubin, {\it C. Mauduit} and {\it A. Sárközy} in [J. Number Theory 106, 56‒69 (2004; Zbl 1049.11089)]: it depends on a polynomial seed $f(x) \in \mathbb{F}_p [x]$. In the same article, the measures $W(E_p)$ and $C_{\ell}(E_p)$ (for some integer $\ell$) of a $Q$-sequence have been bounded. Five basis tests of randomness are well known (see for instance [{\it A. J. Menezes, P. C. van Oorschot} and {\it S. A. Vanstone}, Handbook of applied cryptography. CRC Press Series on Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press (1997; Zbl 0868.94001)]): frequency test, serial test, poker test, runs test, and autocorrelation test. Four of the five statistics used $X_i$, $i \in \{1,2,4,5 \}$ are bounded in the present article, for all binary sequences: the upper bounds use the measures $W(E_N)$ and $C_k(E_N)$, for some integer $k$. Bounds for $Q$-sequences can then be derived. Optimal choices of the parameters $p$ and $f(x)$ are discussed. It is shown that the selected family is reasonably large for cryptographic applications. A running time analysis is performed and the result of a numerical example for a small $p$ and a polynomial $f(x)$ of small degree is given.
Reviewer: Christian Lecot (Le Bourget-du-Lac)