Some highlights of Harald Niederreiter's work

doi:10.1017/CBO9781139696456.002

1 - Some highlights of Harald Niederreiter's work

Published online by Cambridge University Press: 18 December 2014

Gerhard Larcher ,

Friedrich Pillichshammer ,

Arne Winterhof and

Chaoping Xing

Edited by

Gerhard Larcher ,

Friedrich Pillichshammer ,

Arne Winterhof and

Chaoping Xing

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Gerhard Larcher: Affiliation:
Johannes Kepler University Linz
Friedrich Pillichshammer: Affiliation:
Johannes Kepler University Linz
Arne Winterhof: Affiliation:
Austrian Acadamy of Sciences, Linz
Chaoping Xing: Affiliation:
Nanyang Techological University, Singapore
Gerhard Larcher: Affiliation:
Johannes Kepler Universität Linz
Friedrich Pillichshammer: Affiliation:
Johannes Kepler Universität Linz
Arne Winterhof: Affiliation:
Austrian Academy of Sciences, Linz
Chaoping Xing: Affiliation:
Nanyang Technological University, Singapore

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Summary

Dedicated to our teacher, colleague and friend, Harald Niederreiter, on the occasion of his 70th birthday.

Abstract

In this paper we give a short biography of Harald Niederreiter and we spotlight some cornerstones from his wide-ranging work. We focus on his results on uniform distribution, algebraic curves, polynomials and quasi-Monte Carlo methods. In the flavor of Harald's work we also mention some applications including numerical integration, coding theory and cryptography.

A short biography

Harald Niederreiter was born in Vienna in 1944 on June 7 and spent his childhood in Salzburg. In 1963 he returned to Vienna to study at the Department of Mathematics of the University of Vienna, where he finished his PhD thesis entitled “Discrepancy in compact Abelian groups” sub auspiciis praesidentis rei publicae under the supervision of Edmund Hlawka in 1969. From 1969 to 1978 he worked as scientist and professor in the USA at four different institutes: Southern Illinois University, University of Illinois at Urbana-Champaign, Institute for Advanced Study, Princeton, and University of California at Los Angeles. From 1978 to 1981 he was Chair of Pure Mathematics at the University of the West Indies in Kingston (Jamaica). He returned to Austria and served as director of two institutes of the Austrian Academy of Sciences in Vienna, of the Institute for Information Processing until 1999 and then of the Institute of Discrete Mathematics. From 2001 to 2009 he was professor at the National University of Singapore. Since 2009 he has been located at the Johann Radon Institute for Computational and Applied Mathematics in Linz. From 2010 to 2011 he was professor at the King Fahd University of Petroleum and Minerals in Dhahran (Saudi Arabia).

Harald Niederreiter's research areas include numerical analysis, pseudorandom number generation, quasi-Monte Carlo methods, cryptology, finite fields, applied algebra, algorithms, number theory and coding theory. He has published more than 350 research papers and several books, including the following.

Type: Chapter
Information: Applied Algebra and Number Theory , pp. 1 - 21

DOI: https://doi.org/10.1017/CBO9781139696456.002 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2014

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References

[1] J., Dick and H., Niederreiter, Duality for digital sequences. J. Complexity 25, 406–414, 2009.Google Scholar

[2] J., Dick and F., Pillichshammer, Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.

[3] C. S., Ding, H., Niederreiter and C. P., Xing, Some new codes from algebraic curves, IEEE Trans. Inf. Theory 46, 2638–2642, 2000.Google Scholar

[4] J. von zur, Gathen and D., Panario, Factoring polynomials over finite fields: a survey. Computational algebra and number theory (Milwaukee, WI, 1996). J. Symbolic Comput. 31, 3–17, 2001.Google Scholar

[5] F. J., Hickernell and H., Niederreiter, The existence of good extensible rank-1 lattices. J. Complexity 19, 286–300, 2003.Google Scholar

[6] L., Kuipers and H., Niederreiter, Uniform Distribution of Sequences. John Wiley, New York, 1974. Reprint, Dover Publications, Mineola, NY, 2006.

[7] G., Larcher and H., Niederreiter, Kronecker-type sequences and non-Archimedean Diophantine approximations. Acta Arith. 63, 379–396, 1993.Google Scholar

[8] G., Larcher and H., Niederreiter, Generalized (t, s)-sequences, Kronecker-type sequences, and Diophantine approximations of formal Laurent series. Trans. Am. Math. Soc. 347, 2051–2073, 1995.Google Scholar

[9] R., Lidl and H., Niederreiter, Finite Fields, second edition. Encyclopedia of Mathematics and its Applications, volume 20. Cambridge University Press, Cambridge, 1997.

[10] H. B., Mann, The construction of orthogonal Latin squares. Ann. Math. Stat. 13, 418–423, 1942.Google Scholar

[11] C., Mauduit and A., Sárközy, On finite pseudorandom binary sequences. I. Measure of pseudorandomness, the Legendre symbol. Acta Arith. 82, 365–377, 1997.Google Scholar

[12] H., Niederreiter, An application of the Hilbert–Montgomery–Vaughan inequality to the metric theory of uniform distribution mod 1. J. London Math. Soc. 13, 497–506, 1976.Google Scholar

[13] H., Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Am. Math. Soc. 84, 957–1041, 1978.Google Scholar

[14] H., Niederreiter, Dyadic fractions with small partial quotients. Monatsh. Math. 101, 309–315, 1986.Google Scholar

[15] H., Niederreiter, Point sets and sequences with small discrepancy. Monatsh. Math. 104, 273–337, 1987.Google Scholar

[16] H., Niederreiter, Rational functions with partial quotients of small degree in their continued fraction expansion. Monatsh. Math 103, 269–288, 1987.Google Scholar

[17] H., Niederreiter, A short proof for explicit formulas for discrete logarithms in finite fields. Appl. Algebra Eng. Commun. Comput. 1, 55–57, 1990.Google Scholar

[18] H., Niederreiter, Low-discrepancy point sets obtained by digital constructions over finite fields. Czecho. Math. J. 42, 143–166, 1992.Google Scholar

[19] H., Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Series in Applied Mathematics, volume 63. SIAM, Philadelphia, PA, 1992.

[20] H., Niederreiter, A new efficient factorization algorithm for polynomials over small finite fields. Appl. Algebra Eng. Commun. Comput. 4, 81–87, 1993.Google Scholar

[21] H., Niederreiter, Low-discrepancy sequences and non-Archimedean Diophantine approximations. Stud. Sci. Math. Hung. 30, 111–122, 1995.Google Scholar

[22] H., Niederreiter, On the discrepancy of some hybrid sequences. Acta Arith. 138, 373–398, 2009.Google Scholar

[23] H., Niederreiter, A discrepancy bound for hybrid sequences involving digital explicit inversive pseudorandom numbers. Unif. Distrib. Theory 5, 53–63, 2010.Google Scholar

[24] H., Niederreiter, Further discrepancy bounds and an Erdőos–Turán–Koksma inequality for hybrid sequences. Monatsh. Math. 161, 193–222, 2010.Google Scholar

[25] H., Niederreiter, Discrepancy bounds for hybrid sequences involving matrix-method pseudorandom vectors. Publ. Math. Debrecen 79, 589–603, 2011.Google Scholar

[26] H., Niederreiter, Improved discrepancy bounds for hybrid sequences involving Halton sequences. Acta Arith. 155, 71–84, 2012.Google Scholar

[27] H., Niederreiter and F., Özbudak, Constructions of digital nets using global function fields. Acta Arith. 105, 279–302, 2002.Google Scholar

[28] H., Niederreiter and F., Özbudak, Matrix-product constructions of digital nets. Finite Fields Appl. 10, 464–479, 2004.Google Scholar

[29] H., Niederreiter and W., Philipp, Berry–Esseen bounds and a theorem of Erdős and Turán on uniform distribution mod 1. Duke Math. J. 40, 633–649, 1973.Google Scholar

[30] H., Niederreiter and F., Pillichshammer, Construction algorithms for good extensible lattice rules. Construct. Approx. 30, 361–393, 2009.Google Scholar

[31] H., Niederreiter and G., Pirsic, Duality for digital nets and its applications. Acta Arith. 97, 173–182, 2001.Google Scholar

[32] H., Niederreiter and J., Rivat, On the Gowers norm of pseudorandom binary sequences. Bull. Aust. Math. Soc. 79, 259–271, 2009.Google Scholar

[33] H., Niederreiter and K. H., Robinson, Complete mappings of finite fields. J. Aust. Math.Soc.Ser.A 33(2), 197–212, 1982.Google Scholar

[34] H., Niederreiter and I. E., Shparlinski, On the distribution and lattice structure of nonlinear congruential pseudorandom numbers. Finite Fields Appl. 5, 246–253, 1999.Google Scholar

[35] H., Niederreiter and I. E., Shparlinski, On the distribution of inversive congruential pseudorandom numbers in parts of the period. Math. Comp. 70, 1569–1574, 2001.Google Scholar

[36] H., Niederreiter and I., Shparlinski, On the distribution of power residues and primitive elements in some nonlinear recurring sequences. Bull. London Math. Soc. 35, 522–528, 2003.Google Scholar

[37] H., Niederreiter and R. F., Tichy, Solution of a problem of Knuth on complete uniform distribution of sequences. Mathematika 32, 26–32, 1985.Google Scholar

[38] H., Niederreiter and R. F., Tichy, Metric theorems on uniform distribution and approximation theory. Journées Arithmétiques de Besançon (Besançon, 1985), Astérisque No. 147–148, 319–323, 346, 1987.Google Scholar

[39] H., Niederreiter and A., Winterhof, Incomplete character sums and polynomial interpolation of the discrete logarithm. Finite Fields Appl. 8, 184–192, 2002.Google Scholar

[40] H., Niederreiter and A., Winterhof, Multiplicative character sums for nonlinear recurring sequences. Acta Arith. 111, 299–305, 2004.Google Scholar

[41] H., Niederreiter and A., Winterhof, Exponential sums for nonlinear recurring sequences. Finite Fields Appl. 14, 59–64, 2008.Google Scholar

[42] H., Niederreiter and C. P., Xing, Low-discrepancy sequences obtained from algebraic function fields over finite fields. Acta Arith. 72, 281–298, 1995.Google Scholar

[43] H., Niederreiter and C. P., Xing, Low-discrepancy sequences and global function fields with many rational places. Finite Fields Appl. 2, 241–273, 1996.Google Scholar

[44] H., Niederreiter and C. P., Xing, Quasirandom points and global function fields. In: S., Cohen and H., Niederreiter (eds.), Finite Fields and Applications. London Mathematical Society Lecture Note Series, volume 233, pp. 269–296. Cambridge University Press, Cambridge, 1996.

[45] H., Niederreiter and C. P., Xing, Explicit global function fields over the binary field with many rational places. Acta Arith. 75, 383–396, 1996.Google Scholar

[46] H., Niederreiter and C. P., Xing, Cyclotomic function fields, Hilbert class fields and global function fields with many rational places. Acta Arith. 79, 59–76, 1997.Google Scholar

[47] H., Niederreiter and C. P., Xing, Drinfeld modules of rank 1 and algebraic curves with many rational points II. Acta Arith. 81, 81–100, 1997.Google Scholar

[48] H., Niederreiter and C. P., Xing, Global function fields with many rational places over the ternary field. Acta Arith. 83, 65–86, 1998.Google Scholar

[49] H., Niederreiter and C. P., Xing, A general method of constructing global function fields with many rational places. Algorithmic Number Theory (Portland 1998). Lecture Notes in Computer Science, volume 1423, pp. 555–566. Springer, Berlin, 1998.

[50] H., Niederreiter and C. P., Xing, Nets, (t, s)-sequences, and algebraic geometry. Random and Quasi-random Point Sets. Lecture Notes in Statistics, volume 138, pp. 267–302. Springer, New York, 1998.

[51] H., Niederreiter and C. P., Xing, Towers of global function fields with asymptotically many rational places and an improvement on the Gilbert–Varshamov bound. Math. Nachr. 195, 171–186, 1998.Google Scholar

[52] H., Niederreiter and C. P., Xing, Curve sequences with asymptotically many rational points. In: M. D., Fried (ed.), Applications of Curves over Finite Fields. Contemporary Mathematics, volume 245, pp. 3–14. American Mathematical Society, Providence, RI, 1999.

[53] H., Niederreiter and C. P., Xing, Algebraic curves with many rational points over finite fields of characteristic 2. Proc. Number Theory Conference (Zakopane 1997), pp. 359–380. de Gruyter, Berlin, 1999.

[54] H., Niederreiter and C. P., Xing, Rational Points on Curves over Finite Fields. Theory and Applications. London Mathematical Society Lecture Note Series, volume 285. Cambridge University Press, Cambridge, 2001.

[55] H., Niederreiter and C. P., Xing, Constructions of digital nets. Acta Arith. 102, 189–197, 2002.Google Scholar

[56] R., Shaheen and A., Winterhof, Permutations of finite fields for check digit systems. Des. Codes Cryptogr. 57, 361–371, 2010.Google Scholar

[57] A., Winterhof, Recent results on recursive nonlinear pseudorandom number generators. Sequences and their Applications–SETA 2010. Lecture Notes in Computer Science, volume 6338, pp. 113–124. Springer, Berlin, 2010.

[58] A., Winterhof, Generalizations of complete mappings of finite fields and some applications. J. Symbolic Comput. 64, 42–52, 2014.Google Scholar

[59] C. P., Xing and H., Niederreiter, Drinfeld modules of rank 1 and algebraic curves with many rational points. Monatsh. Math. 127, 219–241, 1999.Google Scholar