- Period
- Example
- Compound Inversive Generator
- Example
- Advantages of CIG
- Discrepancy and its boundaries
- Definition
- Notations
- Higher bound
- Lower bound
- See also
- References
- External links
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.
Period
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 {{gaps|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.
Example
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 {{gaps|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.
Example
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.
Definition
For {{mvar|N}} arbitrary points 
the discrepancy is defined by
,
where the supremum is extended over all subintervals {{mvar|J}} of 
, 
is 
times the number of points among
falling into {{mvar|J}} and {{tmath|V(J)}} denotes the {{mvar|s}}-dimensional volume of {{mvar|J}}.
Until now, we had sequences of integers from 0 to {{tmath|T-1}}, in order to have sequences of , one can divide a sequences of integers by its period {{mvar|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 {{mvar|J}} so 
hence
but instead if the sequence is concentrated close to one point then the subinterval {{mvar|J}} is very small 
and 
so 
Then we have from the better and worst case:
.
Notations
Some further notation is necessary. For integers 
and 
let 
be the set of nonzero lattice points 
with 
for 
.
Define
and
for 
. For real 
the abbreviation 
is used, and 
stands for the standard inner product of 
in 
.
Higher bound
Let 
and 
be integers. Let 
with 
for 
.
Then the discrepancy of the points 
satisfies
≤
+
Lower bound
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.
See also
- Pseudorandom number generator
- List of random number generators
- Linear congruential generator
- Generalized inversive congruential pseudorandom numbers
- Naor-Reingold Pseudorandom Function
References
External links
- Inversive Generators at the University of Salzburg.
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