On the linear complexity profile of nonlinear congruential pseudorandom number generators with Rédei functions

Linear complexity and linear complexity profile are important characteristics of a sequence for applications in cryptography and quasi-Monte Carlo methods. The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. We prove lower bounds on the linear complexity profile of nonlinear congruential pseudorandom number generators with Rédei functions which are much stronger than bounds known for general nonlinear congruential pseudorandom number generators.     

Exponential sums of nonlinear congruential pseudorandom number generators with Rédei functions

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. We give new bounds of exponential sums with sequences of iterations of Rédei functions over prime finite fields, which are much stronger than bounds known for general nonlinear congruential pseudorandom number generators.     

On lattice profile of the elliptic curve linear congruential generators

Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order $k$ to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG.     

On the Average Distribution of Inversive Pseudorandom Numbers

The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. The authors have recently introduced a new method for obtaining nontrivial upper bounds on the multidimensional discrepancy of inversive congruential pseudorandom numbers in parts of the period. This method has also been used to study the multidimensional distribution of several other similar families of pseudorandom numbers. Here we apply this method to show that, “on average” over all initial values, much stronger results than those known for “individual” sequences can be obtained.     

Compound nonlinear congruential pseudorandom numbers

The nonlinear congruential method for generating uniform pseudorandom numbers has several very promising properties. However, an implementation in multiprecision of these pseudorandom number generators is usually necessary. In the present paper a compound version of the nonlinear congruential method is introduced, which overcomes this disadvantage. It is shown that the generated sequences have very attractive statistical independence properties. The results that are established are essentially best possible and show that the generated pseudorandom numbers model true random numbers very closely. The method of proof relies heavily on a thorough analysis of exponential sums.     

Distribution of Nonlinear Congruential Pseudorandom Numbers Modulo Almost Squarefree Integers

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new bound on the s-dimensional discrepancy of nonlinear congruential pseudorandom numbers over the residue ring \Bbb Z_M{\Bbb Z}_M modulo M for an “almost squarefree” integer M. It is useful to recall that almost all integers are of this type. Moreover, if the generator is associated with a permutation polynomial over \Bbb Z_M{\Bbb Z}_M we obtain a stronger bound “on average” over all initial values. This bound is new even in the case when M = p is prime.     

On the discrepancy of inversive congruential pseudorandom numbers with prime power modulus,II

One of the alternatives to linear congruential pseudorandom number generators with their known deficiencies is the inversive congruential method with prime power modulus. Recently, it was proved that pairs of inversive congruential pseudorandom numbers have nice statistical independence properties. In the present paper it is shown that a similar result cannot be obtained fork-tuples withk≥3 since their discrepancy is too large. The method of proof relies on the evaluation of certain exponential sums. In view of the present result the inversive congruential method with prime power modulus seems to be not absolutely suitable for generating uniform pseudorandom numbers.     

Distribution of Nonlinear Congruential Pseudorandom Numbers Modulo Almost Squarefree Integers

The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new bound on the s-dimensional discrepancy of nonlinear congruential pseudorandom numbers over the residue ring modulo M for an “almost squarefree” integer M. It is useful to recall that almost all integers are of this type. Moreover, if the generator is associated with a permutation polynomial over we obtain a stronger bound “on average” over all initial values. This bound is new even in the case when M = p is prime.     

On the distribution of inversive congruential pseudorandom numbers in parts of the period

The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present the first nontrivial bounds on the discrepancy of individual sequences of inversive congruential pseudorandom numbers in parts of the period. The proof is based on a new bound for certain incomplete exponential sums.

     

On the Multidimensional Distribution of Inversive Congruential Pseudorandom Numbers in Parts of the Period

 The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present the first nontrivial bounds on the multidimensional discrepancy of individual sequences of inversive congruential pseudorandom numbers in parts of the period. The proof is based on a new bound for certain incomplete exponential sums. (Received 3 December 1998)     

Construction of inversive congruential pseudorandom number generators with maximal period length

The inversive congruential method for generating uniform pseudorandom numbers is a particularly attractive alternative to linear congruential generators with their well-known inherent deficiencies like the unfavourable coarse lattice structure in higher dimensions. In the present paper the modulus in the inversive congruential method is chosen as a power of an arbitrary odd prime. The existence of inversive congruential generators with maximal period length is proved by a new constructive characterization of these generators.     

On the linear complexity profile of some new explicit inversive pseudorandom numbers

Linear complexity and linear complexity profile are interesting characteristics of a sequence for applications in cryptography and Monte-Carlo methods. We introduce some new explicit inversive pseudorandom number generators and prove lower bounds on their linear complexity profile which are close to the best possible.     

Statistical independence of nonlinear congruential pseudorandom numbers

Statistical independence properties of recently proposed nonlinear congruential pseudorandom number generators are analyzed by means of the serial test. The results that are established are essentially best possible. The method relies heavily on bounds for exponential sums.     

On a nonlinear congruential pseudorandom number generator

A nonlinear congruential pseudorandom number generator with modulus is proposed, which may be viewed to comprise both linear as well as inversive congruential generators. The condition for it to generate sequences of maximal period length is obtained. It is akin to the inversive one and bears a remarkable resemblance to the latter.

     

On the discrepancy of inversive congruential pseudorandom numbers with prime power modulus

The inversive congruential method for generating uniform pseudorandom numbers has been introduced recently as an alternative to linear congruential generators with their coarse lattice structure. In the present paper the statistical independence properties of pairs of consecutive pseudorandom numbers obtained from an inversive congruential generator with prime power modulus are analysed by means of the serial test. Upper bounds for the discrepancy of these pairs are established which are essentially best possible. The results show that the inversive congruential method with prime power modulus performs uniformly satisfactorily under the serial test. The methods of proof rely heavily on the evaluation of certain exponential sums which resemble Kloosterman sums.     

On the structure of digital explicit nonlinear and inversive pseudorandom number generators

We analyze the lattice structure and distribution of the digital explicit inversive pseudorandom number generator introduced by Niederreiter and Winterhof as well as of a general digital explicit nonlinear generator. In particular, we extend a lattice test designed for this class of pseudorandom number generators to parts of the period and arbitrary lags and prove that these generators pass this test up to very high dimensions. We also analyze the behavior of digital explicit inversive and nonlinear generators under another very strong lattice test which in its easiest form can be traced back to Marsaglia and provides a complexity measure essentially equivalent to linear complexity.     

On the Distribution of Compound Inversive Congruential Pseudorandom Numbers

 Inversive methods are interesting alternatives to linear methods for pseudorandom number generation. A particularly attractive method is the compound inversive congruential method introduced and analyzed by Huber and Eichenauer-Herrmann. We present the first nontrivial worst-case results on the distribution of sequences of compound inversive congruential pseudorandom numbers in parts of the period. The proofs are based on new bounds for certain exponential sums. (Received 2 March 2000; in revised form 22 November 2000)     

On Marsaglia's lattice test for pseudorandom numbers

Nonlinear recursive congruential pseudorandom number generators with prime modulus and maximal period length are considered. Those generators are characterized which behave optimally with respect to Marsaglia's lattice test. An example for an extremely bad generator with this property is given, which demonstrates the weakness of this test.     

On a new class of inversive pseudorandom numbers for parallelized simulation methods

Inversive methods are attractive alternatives to the linear method for pseudorandom number generation. A particularly attractive method is the digital explicit inversive method recently introduced by the authors. We establish some new results on the statistical properties of parallel streams of pseudorandom numbers generated by this method. In particular, we extend the results of the first author on the statistical properties of pseudorandom numbers generated by the explicit inversive congruential method introduced by Eichenauer-Herrmann. These results demonstrate that the new method is eminently suitable for the generation of parallel streams of pseudorandom numbers with desirable properties.     

Average equidistribution properties of compound nonlinear congruential pseudorandom numbers

The present paper deals with the compound nonlinear congruential method for generating uniform pseudorandom numbers, which has been introduced recently. Equidistribution properties of the generated sequences over parts of the period are studied, based on the discrepancy of the corresponding point sets. Upper and lower bounds for the average value of these discrepancies are established, which are essentially best possible. These results show that the average equidistribution behavior of compound nonlinear congruential pseudorandom numbers fits well the equidistribution properties of true random numbers. The method of proof relies heavily on estimates of the average value of incomplete exponential sums.