Modular constructions of pseudorandom binary sequences with composite moduli. (English) Zbl 1111.11041

C. Mauduit and A. Sarközy [Acta Arith. 82, No. 4, 365–377 (1997; Zbl 0886.11048)] introduced (and analyzed in a series of papers partly with coauthors) three measures of randomness for a finite binary sequence \(E_N\) of length \(N\), the well-distribution measure \(W(E_N)\), the correlation measure \(C_k(E_N)\) of order \(k\), and the normality measure of order \(k\).
In this paper the authors investigate the Jacobi sequences with polynomial argument for a modulus \(m=pq\) with two odd primes \(p<q\) showing that \(W(E_N)\) and \(C_2(E_N)\) are small but \(C_4(E_N)\) is large. They also extend and analyze the constructions of C. Mauduit and the authors [Monatsh. Math. 141, No. 3, 197–208 (2004; Zbl 1110.11024)] and C. Mauduit and A. Sárközy [Acta Math. Hung. 108, No. 3, 239–252 (2005; Zbl 1102.11038)] to composite \(m=pq\).


11K45 Pseudo-random numbers; Monte Carlo methods
11L40 Estimates on character sums
11L07 Estimates on exponential sums
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