Generalized inversive congruential pseudorandom numbers

An approach to nonlinear congruential methods of generating uniform pseudorandom numbers in the interval [0,1) is the Inversive congruential generator with prime modulus. A generalization for arbitrary composite moduli $$m=p_1,\dots p_r$$ with arbitrary distinct primes  $$p_1,\dots ,p_r \ge 5$$ will be present here.

Let $$ \mathbb{Z}_{m} = \{0,1,...,m-1\} $$. For integers $$ a,b \in \mathbb{Z}_{m} $$  with gcd (a,m) = 1 a generalized inversive congruential sequence $$ (y_{n})_{n \geqslant 0} $$ of elements of  $$ \mathbb{Z}_{m} $$ is defined by


 * $$ y_{0} = {\rm seed}$$
 * $$ y_{n+1}\equiv a y_{n}^{\varphi(m)-1} + b \pmod m \text{, }n \geqslant 0 $$

where $$ \varphi(m)=(p_{1}-1)\dots (p_{r}-1)$$ denotes the number of positive integers less than m which are relatively prime to m.

Example
Let take m = 15 = $$ 3 \times 5\, a=2, b=3$$ and $$y_0= 1$$. Hence $$ \varphi (m)= 2 \times 4=8 \,$$ and the sequence $$ (y_{n})_{n \geqslant 0}=(1,5,13,2,4,7,1,\dots )$$  is not maximum.

The result below shows that these sequences are closely related to the following inversive congruential sequence with prime moduli.

For $$1\le i \le r$$ let $$ \mathbb{Z}_{p_{i}}= \{0,1,\dots ,p_{i}-1\}, m_{i}= m / p_{i} $$  and  $$ a_{i} ,b_{i} \in  \mathbb{Z}_{p_{i}} $$ be integers with


 * $$ a\equiv m_{i}^{2} a_{i}\pmod {p_{i}} \;\text{and }\; b\equiv m_{i} b_{i}\pmod {p_{i}} \text{. }$$

Let $$ (y_{n})_{n \geqslant 0} $$ be a sequence of elements of   $$ \mathbb{Z}_{p_{i}}$$, given by


 * $$ y_{n+1}^{(i)}\equiv a_{i} (y_{n}^{(i)})^{p_{i}-2} + b_{i} \pmod {p_{i}}\; \text{, }n \geqslant 0 \;\text {where} \;y_{0}\equiv m_{i} (y_{0}^{(i)})\pmod {p_{i}}\;\text{is assumed. } $$

Theorem 1
Let $$(y_{n}^{(i)})_{n \geqslant 0} $$ for $$1\le i \le r$$ be defined as above. Then
 * $$ y_{n}\equiv m_{1}y_{n}^{(1)} +  m_{2}y_{n}^{(2)} + \dots + m_{r}y_{n}^{(r)}   \pmod m. $$

This theorem shows that an implementation of Generalized Inversive Congruential Generator is possible, where exact integer computations have to be performed only in $$ \mathbb{Z}_{p_{1}},\dots, \mathbb{Z}_{p_{r}}$$ but not in  $$ \mathbb{Z}_{m}. $$

Proof:

First, observe that $$ m_{i}\equiv 0\pmod {p_{j}}, \; \text {for} \; i\ne j, $$ and hence $$ y_{n}\equiv m_{1}y_{n}^{(1)}  +  m_{2}y_{n}^{(2)} + \dots + m_{r}y_{n}^{(r)}   \pmod m $$ if and only if $$y_{n}\equiv m_{i} (y_{n}^{(i)})\pmod {p_{i}}$$, for $$1\le i \le r$$ which will be shown on induction on $$n \geqslant 0 $$.

Recall that $$y_{0}\equiv m_{i} (y_{0}^{(i)})\pmod {p_{i}} $$ is assumed for $$1\le i \le r$$. Now, suppose that $$1\le i \le r$$ and $$y_{n}\equiv m_{i} (y_{n}^{(i)})\pmod {p_{i}}$$ for some integer $$n \geqslant 0 $$. Then straightforward calculations and Fermat's Theorem yield


 * $$ y_{n+1}\equiv a y_{n}^{\varphi(m)-1} + b \equiv m_{i}(a_{i}m_{i}^{\varphi(m)}(y_{n}^{(i)})^{\varphi(m)-1}+ b_{i})\equiv m_{i}(a_{i}(y_{n}^{(i)})^{p_{i}-2} + b_{i})\equiv m_{i}(y_{n+1}^{(i)}) \pmod {p_{i}}$$,

which implies the desired result.

Generalized Inversive Congruential Pseudorandom Numbers are well equidistributed in one dimension. A reliable theoretical approach for assessing their statistical independence properties is based on the discrepancy of s-tuples of pseudorandom numbers.

Discrepancy bounds of the GIC Generator
We use the notation $$D_m ^{s}=D_m (x_0,\dots, x_m-1)$$ where $$ x_n=(x_n, x_n+1 ,\dots , x_n+s-1) $$ $&isin;$ $$ [0,1)^s $$ of Generalized Inversive Congruential Pseudorandom Numbers for $$s\ge 2$$.

Higher bound
 * Let $$s\ge 2$$
 * Then the discrepancy $$ D_m ^s $$ satisfies
 * $$ D_m $$ $$ ^s $$ < $$ m^{-1/2}$$ $&times;$ $$ ( \frac{2}{\pi}$$ $&times;$$$ \log m + \frac{7}{5})^s$$ $&times;$ $$ \textstyle \prod_{i=1}^r (2s-2 + s(p_i)^{-1/2})+ s_{m}^{-1} $$ for any Generalized Inversive Congruential operator.

Lower bound:


 * There exist Generalized Inversive Congruential Generators with
 * $$ D_m $$ $$ ^s $$ $&ge;$ $$ \frac{1}{2(\pi+2)}$$ $&times;$$$ m^{-1/2} $$ :$&times;$ $$ \textstyle \prod_{i=1}^r (\frac {p_{i} - 3}{p_{i} - 1})^{1/2} $$ for all dimension s $
 * &ge;$ 2.

For a fixed number r of prime factors of m, Theorem 2 shows that $$ D_m^{(s)} = O (m^{-1/2} (\log m )^s)$$ for any Generalized Inversive Congruential Sequence. In this case Theorem 3 implies that there exist Generalized Inversive Congruential Generators having a discrepancy $$ D_m^{(s)} $$ which is at least of the order of magnitude $$ m^{-1/2} $$ for all dimension $$s\ge 2$$. However, if m is composed only of small primes, then r can be of an order of magnitude $$ (\log m)/\log\log m $$ and hence $$ \textstyle \prod_{i=1}^r (2s-2 + s(p_i)^{-1/2})= O{(m^\epsilon )} $$ for every $$ \epsilon> 0$$. Therefore, one obtains in the general case $$ D_m^{s}=O(m^{-1/2+\epsilon}) $$ for every $$ \epsilon> 0$$.

Since $$ \textstyle \prod_{i=1}^r ((p_{i} - 3)/(p_{i} - 1))^{1/2} \geqslant 2^{-r/2} $$, similar arguments imply that in the general case the lower bound in Theorem 3 is at least of the order of magnitude $$ m^{-1/2 - \epsilon}$$ for every $$\epsilon> 0$$. It is this range of magnitudes where one also finds the discrepancy of m independent and uniformly distributed random points which almost always has the order of magnitude $$ m^{-1/2} (\log\log m)^{1/2}$$ according to the law of the iterated logarithm for discrepancies. In this sense, Generalized Inversive Congruential Pseudo-random Numbers model true random numbers very closely.