Inversive congruential generator
Inversive congruential generators are a type of nonlinear congruential pseudorandom number generator, which use the modular multiplicative inverse (if it exists) to generate the next number in a sequence. The standard formula for an inversive congruential generator, modulo some prime q is:
seed if if
Such a generator is denoted symbolically as ICG(q,a,c,seed) and is said to be an ICG with parameters q,a,c and seed seed.
The sequence must have after finitely many steps and since the next element depends only on its direct predecessor also etc. The maximum length that the period T for a function modulo q can have is T=q. If the polynomial (polynomial ring over ) is primitive, then the sequence will have the maximum length. Such polynomials are called inversive maximal period (IMP) polynomials. The sufficient condition for maximum sequence period is a proper choice of parameters a and c according to the algorithm described in. Eichenauer-Herrmann, Lehn, Grothe and Niederreiter have shown that inversive congruential generators have good uniformity properties, in particular with regard to lattice structure and serial correlations.
ICG(5,2,3,1) gives the sequence:(1,0,3,2,4,1,.....) (as in , 1 and 4 are their own inverse, 2 is the inverse of 3 and conversely). In this example is irreducible in as neither 0,1,2,3 or 4 are roots, and therefore the period is equal to q=5.In order to show that f is primitive one should show that x is a primitive element of .
Compound Inversive Generator
The construction of a Compound Inversive Generator (CIG) relies on combining two or more congruential inversive generators according to the method described below.
Let be distinct prime integers, each . For each index j,1≤ j ≤ r, let be a sequence of elements of , that is periodic with period length . In other words,.
For each index j, 1≤ j ≤ r, we consider where is the period length of the following sequence .
The sequence of compound pseudorandom numbers is defined as the sum
The compound approach allows combining Inversive Congruential Generators, provided they have full period, in parallel generation systems.
Let and . To simplify, take and . We compute for every 1≤ j≤ 35, then (we have to do the 35 different sums to obtain 0+0 and we begin the same sequence again, the period is ). This method allows obtaining very long period and modular operations may be carried out with relatively small moduli.
Advantages of CIG
The CIG are accepted for practical purposes for a number of reason.
Firstly, binary sequences produced in this way are free of undesirable statistical deviations. Inversive sequences extensively tested with variety of statistical tests remain stable under the variation of parameter.
Secondly, there exists a steady and simple way of parameter choice, based on the Chou algorithm  that guarantees maximum period length.
Thirdly, compound approach has the same properties as single inversive generators  but it also provides period length significantly greater than obtained by a single Inversive Congruential Generator. They seem to be designed for application with multiprocessor parallel hardware platforms.
There exists an algorithm  which allows designing compound generators with predictable period length, predictable linear complexity level, with excellent statistical properties of produced bit streams.
The procedure of designing this complex structure starts with defining finite field of p elements and ends with choosing the parameters a and c for each Inversive Congruential Generator being the component of the compound generator. It means that each generator is associated to a fixed IMP polynomial. Such a condition is sufficient for maximum period of each Inversive Congruential Generator and finally for maximum period of the compound generator. The construction of IMP polynomials is the most efficient approach to find parameters for Inversive Congruential Generator with maximum period length.
Discrepancy and its boundaries
Equidistribution and statistical independence properties of the generated sequences, which are very important for their usability in a stochastic simulation, can be analyzed based on the discrepancy of s-tuples of successive pseudorandom numbers with and respectively.
The discrepancy computes the distance of a generator from a uniform one, a low discrepancy means that the sequence generated can be used for cryptographic purposes and the first aim of the Inversive congruential generator is to provide pseudorandom numbers.
For arbitrary points the discrepancy is defined by , where the supremum is extended over all subintervals of , is times the number of points among falling into J and V(J) denotes the s-dimensional volume of J.
Until now, we had sequences of integers from 0 to T-1, in order to have sequences of , one can divide a sequences of integers by its period T.
From this definition, we can say that if the sequence is perfectly random then its well distributed on the interval then and all points are in J so hence but instead if the sequence is concentrated close to one point then the subinterval J is very small and so Then we have from the better and worst case: .
Some further notation is necessary. For integers and let be the set of nonzero lattice points with for .
for . For real the abbreviation is used, and stands for the standard inner product of .
Let and be integers. Let with for .
Then the discrepancy of the points satisfies
- ≤ +
The discrepancy of arbitrary points satisfies
for any nonzero lattice point , where denotes the number of nonzero coordinates of .
These two theorems show that the CIG is not perfect because the discrepancy is greater strictly than a positive value but also the CIG is not the worst generator as the discrepancy is lower than a value less than 1.
There exist also theorems which bound the average value of the discrepancy for Compound Inversive Generators and also ones which take values such that the discrepancy is bounded by some value depending on the parameters. For more details see the original paper.
- W.S. Chou,On inversive Maximal Period Polynomials over Finite Fields, Applicable Algebra in Engineering, Communication and Computing, No. 4/5, 1995, pp. 245-250.
- J. Eichenauer-Herrmannn. Inversive congruential pseudorandom numbers avoid the planes, Math.Comp., Vol. 56,1991, pp. 297-301.
- J. Eichenauer-Herrmannn, H. Grothe, A. Topuzoglu, On the lattice structure of a nonlinear generator with modulus , J.Comput. Appl. Math., Vol. 31,1990, pp. 81-85.
- J. Eichenauer-Herrmannn, H. Niederreiter, Lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus, Math. Comp., Vol. 58, 1992, pp. 775-779.
- J. Eichenauer-Herrmannn,Statistical independence of a new class of inversive congruential pseudorandom numbers, Math. Comp., Vol 60, 1993, pp. 375-384.
- P. Hellekalek, Inversive pseudorandom number generators:concepts, results and links, Proceedings of the Winter Simulation Conference, 1995, pp 255-262.
- J. Bubicz, J. Stoklosa, Compound Inversive Congruential Generator Design Algorithm, §3 .
- H. Niederreiter, New developments in uniform pseudorandom number and vector generation, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Berlin, 1995.
- J. Eichenauer-Herrmann, F.Emmerich, Compound Inversive Congruential Pseudorandom Numbers: An average-Case Analysis, American Mathematical Society.