Structure of Pseudorandom Numbers Derived from Fermat Quotients

We study the distribution of s-dimensional points of Fermat quotients modulo p with arbitrary lags. If no lags coincide modulo p the same technique as in [21] works. However, there are some interesting twists in the other case. We prove a discrepancy bound which is unconditional for s = 2 and needs restrictions on the lags for s > 2.We apply this bound to derive results on the pseudorandomness of the binary threshold sequence derived from Fermat quotients in terms of bounds on the well-distribution measure and the correlation measure of order 2, both introduced by Mauduit and Sarkozy. We also prove a lower bound on its linear complexity profile. The proofs are based on bounds on exponential sums and earlier relations between discrepancy and both measures above shown by Mauduit, Niederreiter and Sarkozy. Moreover, we analyze the lattice structure of Fermat quotients modulo p with arbitrary lags.

[1]  Harald Niederreiter,et al.  Successive minima profile, lattice profile, and joint linear complexity profile of pseudorandom multisequences , 2008, J. Complex..

[2]  H. Iwaniec,et al.  Analytic Number Theory , 2004 .

[3]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[4]  Shu Lin,et al.  Applied Algebra, Algebraic Algorithms and Error-Correcting Codes , 1999, Lecture Notes in Computer Science.

[5]  András Sárközy,et al.  On finite pseudorandom binary sequences VII: The measures of pseudorandomness , 2002 .

[6]  Arne Winterhof,et al.  Linear complexity profile of binary sequences with small correlation measure , 2006, Period. Math. Hung..

[7]  Arne Winterhof,et al.  Lattice Structure and Linear Complexity Profile of Nonlinear Pseudorandom Number Generators , 2003, Applicable Algebra in Engineering, Communication and Computing.

[8]  D. R. Heath-Brown.  An estimate for Heilbronn's exponential sum , 1996 .

[9]  Wilfried Meidl,et al.  Counting functions and expected values for the lattice profile at n , 2004, Finite Fields Their Appl..

[10]  Igor E. Shparlinski,et al.  Pseudorandomness and Dynamics of Fermat Quotients , 2010, SIAM J. Discret. Math..

[11]  András Sárközy,et al.  On pseudorandom [0, 1 and binary sequences , 2007 .

[12]  Harald Niederreiter,et al.  On the Structure of Inversive Pseudorandom Number Generators , 2007, AAECC.

[13]  Harald Niederreiter,et al.  On the counting function of the lattice profile of periodic sequences , 2007, J. Complex..

[14]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[15]  Harald Niederreiter,et al.  On the correlation of pseudorandom numbers generated by inversive methods , 2008 .

[16]  Robert F. Tichy,et al.  Sequences, Discrepancies and Applications , 1997 .

[17]  G. Marsaglia.  The Structure of Linear Congruential Sequences , 1972 .

[18]  Oscar Moreno,et al.  Exponential sums and Goppa codes: II , 1989, IEEE Trans. Inf. Theory.

[19]  Harald Niederreiter,et al.  Monte Carlo and Quasi-Monte Carlo Methods 2002 , 2004 .

[20]  Arne Winterhof,et al.  Lattice Structure of Nonlinear Pseudorandom Number Generators in Parts of the Period , 2004 .

[21]  Tauno Metsänkylä,et al.  On the p-divisibility of Fermat quotients , 1997, Math. Comput..

[22]  H. Keng,et al.  Applications of number theory to numerical analysis , 1981 .

[23]  Gary L. Mullen,et al.  Finite Fields and Applications , 2007, Student mathematical library.

[24]  Richard Mollin,et al.  Some Conjectures Related to Fermat's Last Theorem , 1990 .

[25]  Harald Niederreiter,et al.  Lattice Structure and Linear Complexity of Nonlinear Pseudorandom Numbers , 2002, Applicable Algebra in Engineering, Communication and Computing.

[26]  András Sárközy,et al.  On finite pseudorandom binary sequences I: Measure of pseudorandomness, the Legendre symbol , 1997 .

[27]  Oscar Moreno,et al.  Exponential sums and Goppa codes. I , 1991 .