On the Carlitz rank of permutations of Fq and pseudorandom sequences

L. Carlitz proved that any permutation polynomial f over a finite field Fq is a composition of linear polynomials and inversions. Accordingly, the minimum number of inversions needed to obtain f is defined to be the Carlitz rank of f by Aksoy et al. The relation of the Carlitz rank of f to other invariants of the polynomial is of interest. Here we give a new lower bound for the Carlitz rank of f in terms of the number of nonzero coefficients of f which holds over any finite field. We also show that this complexity measure can be used to study classes of permutations with uniformly distributed orbits, which, for simplicity, we consider only over prime fields. This new approach enables us to analyze the properties of sequences generated by a large class of permutations of Fp, with the advantage that our bounds for the discrepancy and linear complexity depend on the Carlitz rank, not on the degree. Hence, the problem of the degree growth under iterations, which is the main drawback in all previous approaches, can be avoided.

[1]  Domingo Gómez-Pérez,et al.  Attacking the Pollard Generator , 2006, IEEE Transactions on Information Theory.

[2]  Igor E. Shparlinski,et al.  On the Multidimensional Distribution of Inversive Congruential Pseudorandom Numbers in Parts of the Period , 2000 .

[3]  Hugo Krawczyk.  How to Predict Congruential Generators , 1992, J. Algorithms.

[4]  Harald Niederreiter,et al.  Exponential sums for nonlinear recurring sequences , 2008, Finite Fields Their Appl..

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

[6]  I. Shparlinski.  Cryptographic Applications of Analytic Number Theory , 2003 .

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

[8]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[9]  Alev Topuzoglu,et al.  The Carlitz rank of permutations of finite fields: A survey , 2014, J. Symb. Comput..

[10]  Igor E. Shparlinski,et al.  Predicting the Inversive Generator , 2003, IMACC.

[11]  Igor E. Shparlinski,et al.  On the distribution of inversive congruential pseudorandom numbers in parts of the period , 2001, Math. Comput..

[12]  Arne Winterhof,et al.  Recent Results on Recursive Nonlinear Pseudorandom Number Generators - (Invited Paper) , 2010, SETA.

[13]  Wilfried Meidl,et al.  On the Carlitz rank of permutation polynomials , 2009, Finite Fields Their Appl..

[14]  Jaime Gutierrez,et al.  Inferring sequences produced by a linear congruential generator on elliptic curves missing high-order bits , 2007, Des. Codes Cryptogr..

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  Wilfried Meidl,et al.  Permutations of finite fields with prescribed properties , 2014, J. Comput. Appl. Math..

[17]  Igor E. Shparlinski,et al.  On the Distribution and Lattice Structure of Nonlinear Congruential Pseudorandom Numbers , 1999 .

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

[19]  Igor E. Shparlinski,et al.  Dynamical Systems Generated by Rational Functions , 2003, AAECC.

[20]  Igor E. Shparlinski,et al.  Predicting nonlinear pseudorandom number generators , 2004, Math. Comput..

[21]  Wilfried Meidl,et al.  On the cycle structure of permutation polynomials , 2008, Finite Fields Their Appl..

[22]  Rudolf Lide,et al.  Finite fields , 1983 .

[23]  Igor E. Shparlinski,et al.  On the Distribution of Pseudorandom Numbers and Vectors Generated by Inversive Methods , 2000, Applicable Algebra in Engineering, Communication and Computing.

[24]  Igor E. Shparlinski,et al.  On Stern's Attack Against Secret Truncated Linear Congruential Generators , 2005, ACISP.

[25]  Kenneth G. Paterson,et al.  Permutation Polynomials, de Bruijn Sequences, and Linear Complexity , 1996, J. Comb. Theory, Ser. A.

[26]  Carlos Galindo,et al.  Evaluation codes and plane valuations , 2006, Des. Codes Cryptogr..

[27]  Igor E. Shparlinski,et al.  Cryptographic applications of analytic number theory - complexity lower bounds and pseudorandomness , 2003, Progress in computer science and applied logic.

[28]  Igor E. Shparlinski,et al.  On the linear and nonlinear complexity profile of nonlinear pseudorandom number generators , 2003, IEEE Trans. Inf. Theory.

[29]  H. Niederreiter.  Quasi-Monte Carlo methods and pseudo-random numbers , 1978 .

[30]  L. Carlitz.  Permutations in a finite field , 1953 .