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On the 4-adic complexity of the two-prime quaternary generator

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Abstract

R. Hofer and A. Winterhof proved that the 2-adic complexity of the two-prime (binary) generator of period pq with two odd primes \(p\ne q\) is close to its period and it can attain the maximum in many cases. When the two-prime generator is applied to producing quaternary sequences, we need to determine the 4-adic complexity. It is proved that there are only two possible values of the 4-adic complexity for the two-prime quaternary generator, which are at least \(pq-1-\max \{\log _4(pq^2),\log _4(p^2q)\}\). Examples for primes p and q with \(5\le p, q <10000\) illustrate that the 4-adic complexity only takes one value larger than \(pq-\max \{\log _4(p),\log _4(q)\}\), which is close to its period. So it is good enough to resist the attack of the rational approximation algorithm.

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Notes

  1. Let \(\ell =\gcd \left( 4^p-1, 4^{q}-1 \right) \). Then there exist k, l such that \(kp+lq=1\) and \(4^{kp+lq}\equiv 1\pmod {\ell }\), i.e., \(4\equiv 1\pmod {\ell }\) and \(\ell =3\).

  2. Let \(3^2 \mid (4^p-1)\). That is \(4^p\equiv 1\pmod {9}\). Then p divides the value of the Euler’s totient function \(\varphi (9)=6\) and we have a contradiction since \(p>3\).

  3. If \(\delta \) is of the form \(8\mu +1\), \(8\mu +5\) or \(8\mu +7\), we have from (8)

    $$\begin{aligned} -4(4b+3)^2 +25pq+90-162\lambda \not \equiv \delta (1+2\lambda pq) \pmod {8}. \end{aligned}$$

  4. If \(\delta \) is of the form \(8\mu +1\), \(8\mu +3\) or \(8\mu +5\), we have from (11)

    $$\begin{aligned} -4(4b+3)^2 +9pq-54-162\lambda \not \equiv \delta (1+2\lambda pq) \pmod {8}. \end{aligned}$$

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Acknowledgements

The authors would like to express their gratitude to the referees for their valuable and detailed comments. V. Edemskiy was supported by Russian Science Foundation according to the research Project No. 22-21-00516, https://rscf.ru/en/project/22-21-00516/. Z. Chen was partially supported by the National Natural Science Foundation of China under Grant No. 61772292, and by the Provincial Natural Science Foundation of Fujian, China under Grant No. 2020J01905.

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  1. Zhixiong Chen contributed equally to this work.

Authors and Affiliations

  1. Department of Applied Mathematics and Informatics, Yaroslav-the-Wise Novgorod State University, B. St. Petersburgskaya, 41, Veliky Novgorod, Russia, 173003

    Vladimir Edemskiy

  2. Key Laboratory of Applied Mathematics of Fujian Province University, Putian University, Putian, 351100, Fujian, People’s Republic of China

    Zhixiong Chen

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  1. Vladimir Edemskiy
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Edemskiy, V., Chen, Z. On the 4-adic complexity of the two-prime quaternary generator. J. Appl. Math. Comput. 68, 3565–3585 (2022). https://doi.org/10.1007/s12190-022-01740-z

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  • DOI: https://doi.org/10.1007/s12190-022-01740-z

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