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:

x_0= seed
x_{i+1}\equiv(ax_i^{-1}+ c) \mod q   if x_i\ne 0
x_{i+1}= c if x_i=0

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 (x_n)_{n\geq 0} must have x_i=x_j after finitely many steps and since the next element depends only on its direct predecessor also x_{i+1}=x_{j+1} etc. The maximum length that the period T for a function modulo q can have is T=q. If the polynomial f(x)=x^2-cx-a \in \mathbb F_q[x] (polynomial ring over \mathbb F_q) 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.[1] 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 \in \mathbb F_5 , 1 and 4 are their own inverse, 2 is the inverse of 3 and conversely). In this example f(x)=x^2-3x-2 is irreducible in \mathbb F_5[x] 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 \mathbb F_5[x]/(f).

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 p_1,\dots , p_r be distinct prime integers, each p_j\geq 5. For each index j,1≤ j ≤ r, let (x_n)_{n\geq 0} be a sequence of elements of \in \mathbb F_{p_j} , that is periodic with period length p_j. In other words,\{x_n^{(j)} | 0\leq n \leq p_j\}=\in \mathbb F_{p_j} .

For each index j, 1≤ j ≤ r, we consider T_j=T/p_j where T=p_1\cdots p_r is the period length of the following sequence (x_n)_{n\geq 0}.

The sequence (x_n)_{n\geq 0} of compound pseudorandom numbers is defined as the sum

x_n=T_1x_n^{(1)}+T_2 x_n^{(2)}+\dots +T_rx_n^{(r)} \mod T.

The compound approach allows combining Inversive Congruential Generators, provided they have full period, in parallel generation systems.

Example

Let p_1=5 and p_2 =7 (r =2). To simplify, take (x_n^{(1)})_{n\geq 0} = (0,1,2,3,4,\dots) and (x_n^{(2)})_{n\geq 0} =(0,1,2,3,4,5,6,\dots). We compute for every 1≤ j≤ 35, x_j=7x_j^{(1)}+5x_j^{(2)} \mod 35 then (x_n)_{n\geq 0}=(0, 12, 24, 1, 13, 25, 2, 14, 26, 3, 15, 27, 4, 16, 28, 5, 17, 29, 6, 18, 30, 7, 19, 31, 8, 20, 32, 9, 21, 33, 10, 22, 34, 11, 23) (we have to do the 35 different sums to obtain 0+0 and we begin the same sequence again, the period is 5 \times 7 = 35 \,). 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.[2][3][4]

Secondly, there exists a steady and simple way of parameter choice, based on the Chou algorithm [1] that guarantees maximum period length.

Thirdly, compound approach has the same properties as single inversive generators [5][6] 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 [7] 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[8] 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 s=1 and s=2 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 N arbitrary points {\mathbf t}_1, \dots , {\mathbf t}_{N-1}\in [0,1) the discrepancy is defined by D_N({\mathbf t}_1, \dots , {\mathbf t}_{N-1})={\rm sup}_J|F_N(J)- V(J)|, where the supremum is extended over all subintervals J of [0,1)^s, F_N(J) is N^{-1} times the number of points among {\mathbf t}_1, \dots , {\mathbf t}_{N-1} 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 [0,1)^s, one can divide a sequences of integers by its period T.

From this definition, we can say that if the sequence {\mathbf t}_1, \dots , {\mathbf t}_{N-1} is perfectly random then its well distributed on the interval J=[0,1)^s then V(J)=1 and all points are in J so F_N(J)=N/N=1 hence D_N({\mathbf t}_1, \dots , {\mathbf t}_{N-1})=0 but instead if the sequence is concentrated close to one point then the subinterval J is very small V(j)\approx 0 and F_N(j)\approx N/N\approx 1 so D_N({\mathbf t}_1, \dots , {\mathbf t}_{N-1})=1 Then we have from the better and worst case: 0\leq D_N({\mathbf t}_1, \dots , {\mathbf t}_{N-1})\leq 1.

Notations

Some further notation is necessary. For integers k\geq 1 and q\geq 2 let C_k(q) be the set of nonzero lattice points (h_1,\dots ,h_k)\in Z^k with -q/2< h_j< q/2 for 1\leq j \leq k.

Define

r(h,q)= \begin{cases} q \sin (\pi|h|/q)&\text{for }h \in C_{1}(q)\\ 1 &\text{for }h = 0 \end{cases}

and

r (\mathbf{h},q)=\prod_{j=1}^k r(h_j,q)

for {\mathbf h} =(h_1,\dots ,h_k) \in C_k(q). For real t the abbreviation e(t)={\rm exp}(2\pi\cdot it) is used, and u\cdot v stands for the standard inner product of u,v in R^k.

Higher bound

Let N \geq 1 and q \geq 2 be integers. Let {\mathbf t}_n= y_n/q \in [0,1)^k with y_n \in \{0,1,\dots ,q-1\}^k for 0\leq n< N.

Then the discrepancy of the points {\mathbf t}_0 ,\dots ,{\mathbf t}_{N-1} satisfies

D_N (\mathbf{t}_0,\mathbf{t}_1, \dots ,\mathbf{t}_{N-1}) ≤ \frac kq + \frac 1N \sum_{h \in\C_k(q)}\frac 1{r(\mathbf{h},q)} \Bigg| \sum_{n=0}^{N-1} e(\mathbf{h}\cdot \mathbf{t}_n)\Bigg|

Lower bound

The discrepancy of N arbitrary points \mathbf{t}_1, \dots ,\mathbf{t}_{N-1}\in [0,1)^k satisfies

D_N (\mathbf{t}_0,\mathbf{t}_1, \dots ,\mathbf{t}_{N-1}) ≥ \frac {\pi}{2N((\pi+1)^l -1)\prod_{j=1}^k {\rm max}(1,h_j)}\Bigg|\sum_{n=0}^{N-1} e(\mathbf{h}\cdot \mathbf{t}_n)\Bigg|

for any nonzero lattice point {\mathbf h}=(h_1,\dots ,h_k)\in Z^k, where l denotes the number of nonzero coordinates of {\mathbf h}.

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.[9]

See also

References

  1. ↑ 1.0 1.1 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.
  2. ↑ J. Eichenauer-Herrmannn. Inversive congruential pseudorandom numbers avoid the planes, Math.Comp., Vol. 56,1991, pp. 297-301.
  3. ↑ J. Eichenauer-Herrmannn, H. Grothe, A. Topuzoglu, On the lattice structure of a nonlinear generator with modulus 2^\alpha, J.Comput. Appl. Math., Vol. 31,1990, pp. 81-85.
  4. ↑ 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.
  5. ↑ J. Eichenauer-Herrmannn,Statistical independence of a new class of inversive congruential pseudorandom numbers, Math. Comp., Vol 60, 1993, pp. 375-384.
  6. ↑ P. Hellekalek, Inversive pseudorandom number generators:concepts, results and links, Proceedings of the Winter Simulation Conference, 1995, pp 255-262.
  7. ↑ J. Bubicz, J. Stoklosa, Compound Inversive Congruential Generator Design Algorithm, §3 .
  8. ↑ H. Niederreiter, New developments in uniform pseudorandom number and vector generation, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Berlin, 1995.
  9. ↑ J. Eichenauer-Herrmann, F.Emmerich, Compound Inversive Congruential Pseudorandom Numbers: An average-Case Analysis, American Mathematical Society.

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