Skip to main content
SpringerLink
Log in

Pseudorandom sequences derived from automatic sequences

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Many automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, have some desirable features of pseudorandomness such as a large linear complexity and a small well-distribution measure. However, they also have some undesirable properties in view of certain applications. For example, the majority of possible binary patterns never appears in automatic sequences and their correlation measure of order 2 is extremely large. Certain subsequences, such as automatic sequences along squares, may keep the good properties of the original sequence but avoid the bad ones. In this survey we investigate properties of pseudorandomness and non-randomness of automatic sequences and their subsequences and present results on their behaviour under several measures of pseudorandomness including linear complexity, correlation measure of order k, expansion complexity and normality. We also mention some analogs for finite fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Notes

  1. Apwenian sequences are named after the authors of [5].

  2. Two integers k and ℓ are multiplicatively dependent if kr = ℓs for some positive integers r and s. Otherwise they are multiplicatively independent.

  3. For a prime power q we denote the finite field of size q by \(\mathbb {F}_{q}\).

  4. We denote by \(\mathbb {F}_{q} \llbracket x \rrbracket \) the ring of formal power series over \(\mathbb {F}_{q}\).

  5. f(k) = o(g(k)) is equivalent to \(f(k)/ g(k)\rightarrow 0\) as \(k\rightarrow \infty \).

  6. Note that the term balanced is used with a different meaning in combinatorics on words, see for example [8, Definition 10.5.4].

  7. f(k) = O(g(k)) is equivalent to |f(k)|≤ cg(k) for some constant c > 0.

  8. f(k) = Θ(g(k)) is equivalent to c1g(k) ≤ f(k) ≤ c2g(k) for some constants c2 ≥ c1 > 0.

  9. \(\overline {\mathbb {F}_{p}}=\bigcup _{n=1}^{\infty } \mathbb {F}_{p^{n}}\) denotes the algebraic closure of \(\mathbb {F}_{p}\).

References

  1. Aistleitner, C.: On the limit distribution of the well-distribution measure of random binary sequences. J. Thé,or. Nombres Bordeaux 25(2), 245–259 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allouche, J.-P.: Somme des chiffres et transcendance. Bull. Soc. Math France 110 3, 279–285 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Allouche, J.-P.: On a Golay-Shapiro-like sequence. Unif. Distrib. Theory 11(2), 205–210 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allouche, J.-P., Han, G.-N., Niederreiter, H.: Perfect linear complexity profile and apwenian sequences. Finite Fields Appl. 68(101761), 13 (2020)

    MathSciNet  MATH  Google Scholar 

  5. Allouche, J.-P., Peyriére, J., Wen, Z.-X., Wen, Z.-Y.: Hankel determinants of the Thue-Morse sequence. Ann. Inst. Fourier (Grenoble) 48(1), 1–27 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Allouche, J.-P., Salon, O.: Sous-suites polynomiales de certaines suites automatiques. J. Thé,or. Nombres Bordeaux 5(1), 111–121 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Allouche, J.-P., Shallit, J.: The Ubiquitous Prouhet-Thue-Morse Sequence. Sequences and Their Applications (Singapore, 1998), 1–16, Springer Ser. Discrete Math. Theor. Comput. Sci. Springer, London (1999)

    Google Scholar 

  8. Allouche, J.-P., Shallit, J.: Automatic Sequences. Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  9. Allouche, J.-P., Shallit, J., Yassawi, R.: How to prove that a sequence is not automatic. To appear in Expositiones Mathematicae, online available https://doi.org/10.1016/j.exmath.2021.08.001 (2021)

  10. Alon, N., Kohayakawa, Y., Mauduit, C., Moreira, C.G., Rödl, V.: Measures of pseudorandomness for finite sequences: typical values. Proc. Lond. Math. Soc. (3) 95(3), 778–812 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Blumer, A., Blumer, J., Ehrenfeucht, A., Haussler, D., McConnell, R.: Linear size finite automata for the set of all subwords of a word: an outline of results. Bul. Eur. Assoc. Theor. Comp. Sci. 21, 12–20 (1983)

    MATH  Google Scholar 

  12. Bourgain, J.: Prescribing the binary digits of primes. Israel J. Math. 194(2), 935–955 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bourgain, J.: Prescribing the binary digits of primes, II. Israel J. Math. 206(1), 165–182 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brlek, S.: Enumeration of factors in the Thue-Morse word. First Montreal Conference on Combinatorics and Computer Science, 1987. Discrete Appl Math. 24 (1-3), 83–96 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Car, M.: Distribution des polynômes irréductibles dans fq[t]. Acta Arith. 88(2), 141–153 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cassaigne, J., Ferenczi, S., Mauduit, C., Rivat, J., Sárközy, A.: On finite pseudorandom binary sequences. III. The Liouville function. I. Acta Arith. 87(4), 367–390 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chan, L., Grimm, U.: Spectrum of a Rudin-Shapiro-like sequence. Adv. in Appl. Math. 87, 16–23 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Christol, G., presque, Ensembles: Périodiques k-reconnaissables. Theoret. Comput. Sci. 9(1), 141–145 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Christol, G., Kamae, T., Mendès France, M., Rauzy, G. : Suites algébriques, automates et substitutions. Bull. Soc. Math. France 108(4), 401–419 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cox, D.A., Little, J., O’Shea, D.: Ideals, varieties and Algorithms. Undergraduate Texts in Mathematics. Springer, Cham, fourth edition. An introduction to computational algebraic geometry and commutative algebra (2015)

  21. Dartyge, C., Mauduit, C., Sárközy, A.: Polynomial values and generators with missing digits in finite fields. Funct. Approx Comment. Math. 52(1), 65–74 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dartyge, C., Mérai, L., Winterhof, A.: On the distribution of the Rudin-Shapiro function for finite fields. Proc. Amer. Math. Soc. 149(12), 5013–5023 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dartyge, C., Sárközy, A.: The sum of digits function in finite fields. Proc. Amer. Math. Soc. 141(12), 4119–4124 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Deligne, P.: La conjecture de Weil. Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  25. de Luca, A., Varricchio, S.: Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups. Theoret Comput. Sci. 63(3), 333–348 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Diem, C.: On the use of expansion series for stream ciphers. LMS. J. Comput. Math. 15, 326–340 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Dietmann, R., Elsholtz, C., Shparlinski, I.E.: Prescribing the binary digits of squarefree numbers and quadratic residues. Trans. Amer. Math. Soc. 369 (12), 8369–8388 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dorfer, G.: Lattice profile and linear complexity profile of pseudorandom number sequences. Finite fields and applications, 69–78, Lecture Notes in Comput. Sci Springer, Berlin (2948)

  29. Dorfer, G., Meidl, W., Winterhof, A.: Counting functions and expected values for the lattice profile at n. Finite Fields Appl. 10(4), 636–652 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dorfer, G., Winterhof, A.: Lattice structure and linear complexity profile of nonlinear pseudorandom number generators. Appl. Algebra Engrg. Comm Comput. 13(6), 499–508 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dorfer, G., Winterhof, A.: Lattice Structure of Nonlinear Pseudorandom Number Generators in Parts of the Period. Monte Carlo and quasi-Monte-Methods 2002, 199–211. Springer, Berlin (2004)

    MATH  Google Scholar 

  32. Drmota, M.: Subsequences of Automatic Sequences and Uniform Distribution. Uniform Distribution and quasi-Monte Carlo Methods, 87–104, Radon Ser. Comput. Appl Math., vol. 15. De Gruyter, Berlin (2014)

    Google Scholar 

  33. Drmota, M., Mauduit, C., Rivat, J.: Normality along squares. J. Eur. Math. Soc. 21(2), 507–548 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  34. Drmota, M., Müllner, C., Spiegelhofer, L.: Primes as sums of Fibonacci numbers. Preprint 2021.2109.04068 (2021)

  35. Dupuy, T., Weirich, D.E.: Bits of 3n in binary, Wieferich primes and a conjecture of Erdös. J. Number Theory 158, 268–280 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Everest, G., van der Poorten, A., Shparlinski, I., Ward, T.: Recurrence Sequences Mathematical Surveys and Monographs, 104. American Mathematical Society, Providence, RI (2003)

  37. Fogg, N.P.: Substitutions in Dynamics, Arithmetics and Combinatorics. Edited by V. BerthÉ, S. Ferenczi, C. Mauduit and A. Siegel Lecture Notes in Mathematics, vol. 1794. Springer-Verlag, Berlin (2002)

    Google Scholar 

  38. Fouvry, E., Mauduit, C.: Sommes des chiffres et nombres presque premiers. Math. Ann. 305, 571–599 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. Gabdullin, M.R.: On the squares in the set of elements of a finite field with constraints on the coefficients of its basis expansion. (Russian) Mat. Zametki 100 (2016), no. 6, 807–824; translation in Math. Notes 101, no. 1–2, 234–249 (2017)

  40. Gao, Z., Kuttner, S., Wang, Q.: On enumeration of irreducible polynomials and related objects over a finite field with respect to their trace and norm. Finite Fields Appl. 69(101770), 25 (2021)

    MathSciNet  MATH  Google Scholar 

  41. Gel’fond, A.O.: Sur les nombres qui ont des propriétés additives et multiplicatives données. (French) Acta Arith. 13 , 259–265 (1967)

  42. Gómez-Pérez, D., Mérai, L.: Algebraic dependence in generating functions and expansion complexity. Adv. Math. Commun. 14(2), 307–318 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  43. Gómez-Pérez, D., Mérai, L., Niederreiter, H.: On the expansion complexity of sequences over finite fields. IEEE Trans. Inform. Theory 64(6), 4228–4232 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  44. Granger, R.: On the enumeration of irreducible polynomials over GF(q) with prescribed coefficients. Finite Fields Appl. 57, 156–229 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gyarmati, K.: Measures of Pseudorandomness. Finite Fields and Their Applications. 43–64, Radon Ser. Comput. Appl Math., vol. 11. De Gruyter, Berlin (2013)

    Google Scholar 

  46. Ha, J.: Irreducible polynomials with several prescribed coefficients. Finite Fields Appl. 40, 10–25 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Hofer, R., Winterhof, A., complexity, Linear: Expansion Complexity of Some Number Theoretic Sequences. Arithmetic of Finite Fields, 67–74 Lecture Notes in Comput Sci., vol. 10064. Springer, Cham (2016)

    Google Scholar 

  48. Hooley, C.: On the number of points on a complete intersection over a finite field. With an appendix by Nicholas M. Katz. J. Number Theory 38(3), 338–358 (1991)

    Article  MATH  Google Scholar 

  49. Jamet, D., Popoli, P., Stoll, T.: Maximum order complexity of the sum of digits function in Zeckendorf base and polynomial subsequences. Cryptogr. Commun. 13(5), 791–814 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  50. Jansen, C.J.A.: Investigations on nonlinear streamcipher systems: Construction and evaluation methods. Thesis (Dr.)–Technische Universiteit Delft (The Netherlands). Proquest LLC, Ann Arbor, MI, p 195 (1989)

  51. Jansen, C.J.A., Boekee, D.E.: The Shortest Feedback Shift Register that Can Generate a Given Sequence. Advances in Cryptology—CRYPTO ’89 (Santa Barbara, CA, 1989), 90–99, Lecture Notes in Comput Sci., vol. 435. Springer, New York (1990)

    Google Scholar 

  52. Kaneko, H., Stoll, T.: On subwords in the base-q expansion of polynomial and exponential functions. Integers 18A Paper A11, 11 (2018)

    MathSciNet  MATH  Google Scholar 

  53. Lafrance, P., Rampersad, N., Yee, R.: Some properties of a Rudin-Shapiro-like sequence. Adv. in Appl. Math. 63, 19–40 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Lagarias, J.: Ternary expansions of powers of 2. J. Lond. Math. Soc. (2) 79(3), 562–588 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  55. L’Ecuyer, P., Simard, R.: Testu01: A C Library for Empirical Testing of Random Number Generators. ACM Transactions on Mathematical Software. Vol. 33, article 22 (2007)

  56. Lidl, R., Niederreiter, H.: Finite Fields. Second Edition Encyclopedia of Mathematics and Its Applications, vol. 20. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  57. Makhul, M., Winterhof, A.: Normality of the Thue-Morse function for finite fields along polynomial values. Preprint 2021. https://arxiv.org/abs/2106.12218 (2021)

  58. Marcovici, I., Stoll, T., Tahay, P.-A.: Discrete correlations of order 2 of generalized Golay-Shapiro sequences: A combinatorial approach. Integers 21, Paper No. A45, 21 pp (2021)

  59. Mattheus, S.: Trace of products in finite fields from a combinatorial point of view. SIAM J. Discrete Math. 33(4), 2126–2139 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  60. Mauduit, C., Rivat, J.: La somme des chiffres des carrés. Acta Math. 203(1), 107–148 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  61. Mauduit, C., Rivat, J.: Sur un probléme de gelfond: la somme des chiffres des nombres premiers. Ann. of Math. (2) 171(3), 1591–1646 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  62. Mauduit, C., Rivat, J.: Prime numbers along Rudin-Shapiro sequences. J. Eur. Math. Soc. (JEMS) 17(10), 2595–2642 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. Mauduit, C., Rivat, J.: Rudin-shapiro sequences along squares. Trans. Amer. Math. Soc. 370(11), 7899–7921 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  64. Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82(4), 365–377 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  65. Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences. II. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences, a further construction. J. Number Theory 73(2), 256–276 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  66. Mauduit, C., Sárközy, A.: On the measures of pseudorandomness of binary sequences. Discrete Math. 271, no. 1–3, 195–207 (2003)

  67. Mérai, L., Niederreiter, H., Winterhof, A.: Expansion complexity and linear complexity of sequences over finite fields Cryptogr. Commun. 9(4), 501–509 (2017)

    MathSciNet  MATH  Google Scholar 

  68. Mérai, L., Rivat, J., Sárközy, A.: The Measures of Pseudorandomness and the NIST Tests. Number-theoretic Methods in Cryptology, 197–216, Lecture Notes in Comput Sci., 10737, Springer, Cham (2018)

  69. Mérai, L., Winterhof, A.: On the pseudorandomness of automatic sequences. Cryptogr. Commun. 10(6), 1013–1022 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  70. Mérai, L., Winterhof, A.: On the N th linear complexity of automatic sequences. J. Number Theory 187, 415–429 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  71. Moshe, Y.: On the subword complexity of Thue-Morse polynomial extractions. Theoret. Comput. Sci. 389(1–2), 318–329 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  72. Mullen, G.L., Panario, D. (eds.): Handbook of Finite Fields. Discrete Mathematics and Its Applications (Boca Raton). CRC Press, Boca Raton, FL (2013)

  73. Müllner, C.: The Rudin-Shapiro sequence and similar sequences are normal along squares. Canad. J. Math. 70(5), 1096–1129 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  74. Müllner, C., Spiegelhofer, L.: Normality of the Thue-Morse sequence along Piatetski-Shapiro sequences, II. Israel. J. Math. 220(2), 691–738 (2017)

    MathSciNet  MATH  Google Scholar 

  75. Niederreiter, H.: Sequences with Almost Perfect Linear Complexity Profile. Advances in Cryptology-EUROCRYPT ’87 (D. Chaum and W. L. Price, Eds.), Lecture Notes in Computer Science, Vol. 304, Pp. 37–51. Springer-Verlag, Berlin/Heidelberg/New York (1988)

    Google Scholar 

  76. Niederreiter, H.: The Probabilistic Theory of Linear Complexity. Advances in Cryptology – EUROCRYPT ’88 (C. G. Günther, Ed.) Lecture Notes in Computer Science, Vol. 330, Pp. 191–209. Springer, Berlin (1988)

    Google Scholar 

  77. Niederreiter, H.: Linear complexity and related complexity measures for sequences. Progress in Cryptology—INDOCRYPT 2003, 1–17, Lecture Notes in Comput. Sci. 2904, Springer, Berlin (2003)

  78. Niederreiter, H., Xing, C.: Sequences with high nonlinear complexity. IEEE Trans. Inform. Theory 60(10), 6696–6701 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  79. Niederreiter, H., Winterhof, A.: Applied number theory. Springer Cham (2015)

  80. Ostafe, A.: Polynomial values in affine subspaces of finite fields. J. Anal. Math. 138(1), 49–81 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  81. Pollack, P.: Irreducible polynomials with several prescribed coefficients. Finite Fields Appl. 22, 70–78 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  82. Popoli, P.: On the maximum order complexity of Thue-Morse and Rudin-Shapiro sequences along polynomial values. Unif. Distrib. Theory 15(2), 9–22 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  83. Porritt, S.: Irreducible polynomials over a finite field with restricted coefficients. Canad. Math. Bull. 2, 429–439 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  84. Rudin, W.: Some theorems on Fourier coefficients. Proc. Amer. Math. Soc. 10, 855–859 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  85. Rukhin, A., et al.: NIST Special Publication 800-22, Revision 1.a, A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications, https://www.nist.gov/publications/statistical-test-suite-random-and-pseudorandom-number-generators-cryptographic(2021)

  86. Schmidt, K.-U.: The correlation measures of finite sequences: limiting distributions and minimum values. Trans. Amer. Math. Soc. 369(1), 429–446 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  87. Shapiro, H.S.: Extremal Problems for Polynomials and Power Series. Master’s thesis, MIT (1952)

  88. Shparlinski, I.: Cryptographic Applications of Analytic Number Theory. Complexity Lower Bounds and Pseudorandomness Progress in Computer Science and Applied Logic, vol. 22. Basel, Birkhäuser Verlag (2003)

    Google Scholar 

  89. Spiegelhofer, L.: Normality of the Thue-Morse sequence along Piatetski-Shapiro sequences. Q. J. Math. 66(4), 1127–1138 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  90. Spiegelhofer, L.: The level of distribution of the Thue-Morse sequence. Compos. Math. 156(12), 2560–2587 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  91. Stoll, T.: The sum of digits of polynomial values in arithmetic progressions. Funct. Approx. Comment. Math. 47(2), 233–239 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  92. Stoll, T.: Combinatorial constructions for the Zeckendorf sum of digits of polynomial values. Ramanujan J. 32(2), 227–243 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  93. Stoll, T.: On digital blocks of polynomial values and extractions in the Rudin-Shapiro sequence. RAIRO Theor. Inform. Appl. 50(1), 93–99 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  94. Sun, Z., Winterhof, A.: On the maximum order complexity of subsequences of the Thue-Morse and Rudin-Shapiro sequence along squares. Int. J. Comput. Math. Comput. Syst. Theory 4(1), 30–36 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  95. Sun, Z., Winterhof, A.: On the maximum order complexity of the Thue-Morse and Rudin-Shapiro sequence. Unif. Distrib. Theory 14(2), 33–42 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  96. Sun, Z., Zeng, X., Lin, D.: On the N th maximum order complexity and the expansion complexity of a Rudin-Shapiro-like sequence. Cryptogr. Commun. 12(3), 415–426 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  97. Swaenepoel, C.: Trace of products in finite fields. Finite Fields Appl. 51, 93–129 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  98. Swaenepoel, C.: On the sum of digits of special sequences in finite fields. Monatsh. Math. 187(4), 705–728 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  99. Swaenepoel, C.: Prescribing digits in finite fields. J. Number Theory 189, 97–114 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  100. Topuzoğlu, A., Winterhof, A.: Pseudorandom sequences. Topics in geometry, coding theory and cryptography, 135–166, Algebr Appl., vol. 6. Springer, Dordrecht (2007)

    Google Scholar 

  101. Tuxanidy, A., Wang, Q.: Irreducible polynomials with prescribed sums of coefficients. Preprint 2016. https://arxiv.org/abs/1605.00351 (2016)

  102. Winterhof, A.: Linear Complexity and Related Complexity Measures. Selected Topics in Information and Coding Theory, 3–40, Ser Coding Theory Cryptol., vol. 7. World Sci. Publ., Hackensack, NJ (2010)

    Google Scholar 

  103. Winterhof, A.: Recent Results on Recursive Nonlinear Pseudorandom Number Generators (Invited Paper). Sequences and Their Applications-SETA 2010, 113–124, Lecture Notes in Comput Sci., vol. 6338. Springer, Berlin (2010)

    Google Scholar 

  104. Xing, C., Lam, K.: Sequences with almost perfect linear complexity profiles and curves over finite fields. IEEE Trans. Inform. Theory, pp 1267–1270 (1999)

Download references

Acknowledgment

The authors were supported by the Austrian Science Fund FWF grants P 30405 and P 31762. They wish to thank Jean-Paul Allouche, Harald Niederreiter, Igor Shparlinski, Cathy Swaenepoel, Thomas Stoll and Steven Wang for very useful discussions as well as the anonymous referees and the associate editor Daniel Katz for their concise and very helpful reports.

Author information

Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences, Altenbergerstr. 69, 4040, Linz, Austria

    László Mérai & Arne Winterhof

Authors
  1. László Mérai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arne Winterhof
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to László Mérai.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article belongs to the Topical Collection: Surveys (invitation only)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mérai, L., Winterhof, A. Pseudorandom sequences derived from automatic sequences. Cryptogr. Commun. 14, 783–815 (2022). https://doi.org/10.1007/s12095-022-00556-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-022-00556-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation