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More constructions of pseudorandom lattices of \(k\) symbols

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Abstract

We define the pseudorandom lattice of \(k\) symbols, and present some new constructions of families of such lattices, which generalize several previous constructions for the pseudorandom binary lattices. These lattices can also be regarded as a high dimensional analogue of some pseudorandom sequences of \(k\) symbols.

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References

  1. Ahlswede, R., Mauduit, C., Sárközy, A.: Large families of pseudorandom sequences of \(k\) symbols and their complexity. I, General Theory of Information Transfer and Combinatorics, Lecture Notes in Comput. Sci., vol. 4123, pp. 293–307. Springer, Berlin (2006)

  2. Ahlswede, R., Mauduit, C., Sárközy, A.: Large families of pseudorandom sequences of \(k\) symbols and their complexity. II, General Theory of Information Transfer and Combinatorics, Lecture Notes in Comput. Sci., vol. 4123, pp. 308–325, Springer, Berlin (2006)

  3. Bérczi, G.: On finite pseudorandom sequences of \(k\) symbols. Period. Math. Hungar. 47(1–2), 29–44 (2003)

    MATH  MathSciNet  Google Scholar 

  4. Eichenauer-Herrmann, J., Niederreiter, H.: Bounds for exponential sums and their applications to pseudorandom numbers. Acta Arith. 67(3), 269–281 (1994)

    MATH  MathSciNet  Google Scholar 

  5. Gomez, D., Winterhof, A.: Multiplicative character sums of Fermat quotients and pseudorandom sequences. Period. Math. Hungar. 64(2), 161–168 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hubert, P., Mauduit, C., Sárközy, A.: On pseudorandom binary lattices. Acta Arith. 125(1), 51–62 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Liu, H.: A family of pseudorandom binary sequences constructed by the multiplicative inverse. Acta Arith. 130(2), 167–180 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Liu, H., Yang, C.: On a problem of D. H. Lehmer and pseudorandom binary sequences. Bull. Braz. Math. Soc. (N.S.) 39(3), 387–399 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu, H.: A large family of pseudorandom binary lattices. Proc. Am. Math. Soc. 137(3), 793–803 (2009)

    Article  MATH  Google Scholar 

  10. Liu, H.: Large families of pseudorandom binary sequences and lattices by using the multiplicative inverse. Acta Arith. 159(2), 123–131 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mak, K.-H.: More constructions of pseudorandom sequences of \(k\) symbols. Finite Fields Appl. 25, 222–233 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mauduit, C., Rivat, J., Sárközy, A.: Construction of pseudorandom binary sequences using additive characters. Monatsh. Math. 141(3), 197–208 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  14. Mauduit, C., Sárközy, A.: On finite pseudorandom sequences of \(k\) symbols. Indag. Math. (N.S.) 13, 89–101 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mauduit, C., Sárközy, A.: Construction of pseudorandom binary sequences by using the multiplicative inverse. Acta Math. Hungar. 108(3), 239–252 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mauduit, C., Sárközy, A.: On large families of pseudorandom binary lattices. Unif. Distrib. Theory 2(1), 23–37 (2007)

    MATH  MathSciNet  Google Scholar 

  17. Mauduit, C., Sárközy, A.: Construction of pseudorandom binary lattices by using the multiplicative inverse. Monatsh. Math. 153(3), 217–231 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Mérai, L.: Construction of pseudorandom binary lattices based on multiplicative characters. Period. Math. Hungar. 59(1), 43–51 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mérai, L.: A construction of pseudorandom binary sequences using rational functions. Unif. Distrib. Theory 4(1), 35–49 (2009)

    MATH  MathSciNet  Google Scholar 

  20. Mérai, L.: On finite pseudorandom lattices of \(k\) symbols. Monatsh. Math. 161(2), 173–191 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mérai, L.: Construction of pseudorandom binary lattices using elliptic curves. Proc. Am. Math. Soc. 139(2), 407–420 (2011)

    Article  MATH  Google Scholar 

  22. Mérai, L.: Construction of pseudorandom binary sequences over elliptic curves using multiplicative characters. Publ. Math. Debrecen 80(1–2), 199–213 (2012)

    MATH  MathSciNet  Google Scholar 

  23. Sárközy, A.: On finite pseudorandom binary sequences and their applications in cryptography. Tatra Mt. Math. Publ. 37, 123–136 (2007)

    MATH  MathSciNet  Google Scholar 

  24. Winterhof, A.: Some estimates for character sums and applications. Des. Codes Cryptogr. 22(2), 123–131 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wu, C., Weng, X., Chen, Z.: Construction of \(k\)-ary pseudorandom elliptic curve sequences. Wuhan Univ. J. Nat. Sci. 16(5), 452–456 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  26. Zhang, W.: On a problem of D. H. Lehmer and its generalization. Compositio Math. 86(3), 307–316 (1993)

    MATH  MathSciNet  Google Scholar 

  27. Zhang, W.: On the difference between a D. H. Lehmer number and its inverse modulo \(q\). Acta Arith. 68(3), 255–263 (1994)

    MATH  MathSciNet  Google Scholar 

  28. Zhang, W.: A problem of D. H. Lehmer and its generalization. II. Compositio Math. 91(1), 47–56 (1994)

    MATH  MathSciNet  Google Scholar 

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Correspondence to Kit-Ho Mak.

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Communicated by J. Schoißengeier.

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Mak, KH. More constructions of pseudorandom lattices of \(k\) symbols. Monatsh Math 177, 307–323 (2015). https://doi.org/10.1007/s00605-014-0663-x

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  • DOI: https://doi.org/10.1007/s00605-014-0663-x

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