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

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

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

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

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

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

Let ${\displaystyle (y_{n}^{(i)})_{n\geqslant 0}}$  for ${\displaystyle 1\leq i\leq r}$  be defined as above. Then

${\displaystyle y_{n}\equiv m_{1}y_{n}^{(1)}+m_{2}y_{n}^{(2)}+\dots +m_{r}y_{n}^{(r)}{\pmod {m}}.}$ 

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

Proof:

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

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

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

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.

We use the notation ${\displaystyle D_{m}^{s}=D_{m}(x_{0},\dots ,x_{m}-1)}$  where ${\displaystyle x_{n}=(x_{n},x_{n}+1,\dots ,x_{n}+s-1)}$  ∈ ${\displaystyle [0,1)^{s}}$  of Generalized Inversive Congruential Pseudorandom Numbers for ${\displaystyle s\geq 2}$ .

Higher bound

Let ${\displaystyle s\geq 2}$ 

Then the discrepancy ${\displaystyle D_{m}^{s}}$  satisfies

${\displaystyle D_{m}}$  ${\displaystyle ^{s}}$  < ${\displaystyle m^{-1/2}}$  × ${\displaystyle ({\frac {2}{\pi }}}$  × ${\displaystyle \log m+{\frac {7}{5}})^{s}}$  × ${\displaystyle \textstyle \prod _{i=1}^{r}(2s-2+s(p_{i})^{-1/2})+s_{m}^{-1}}$  for any Generalized Inversive Congruential operator.

Lower bound:

There exist Generalized Inversive Congruential Generators with

${\displaystyle D_{m}}$  ${\displaystyle ^{s}}$  ≥ ${\displaystyle {\frac {1}{2(\pi +2)}}}$  × ${\displaystyle m^{-1/2}}$  : × ${\displaystyle \textstyle \prod _{i=1}^{r}({\frac {p_{i}-3}{p_{i}-1}})^{1/2}}$  for all dimension s  :≥ 2.

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

Since ${\displaystyle \textstyle \prod _{i=1}^{r}((p_{i}-3)/(p_{i}-1))^{1/2}\geqslant 2^{-r/2}}$ , similar arguments imply that in the general case the lower bound in Theorem 3 is at least of the order of magnitude ${\displaystyle m^{-1/2-\epsilon }}$  for every ${\displaystyle \epsilon >0}$ . 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 ${\displaystyle m^{-1/2}(\log \log m)^{1/2}}$  according to the law of the iterated logarithm for discrepancies.^[2] In this sense, Generalized Inversive Congruential Pseudo-random Numbers model true random numbers very closely.

• Pseudorandom number generator
• List of random number generators
• Linear congruential generator
• Inversive congruential generator
• Naor-Reingold Pseudorandom Function

• Eichenauer-Herrmann, Jürgen (1994), "On Generalized Inversive Congruential Pseudorandom Numbers", Mathematics of Computation, 63 (207) (first ed.), American Mathematical Society: 293–299, doi:10.1090/S0025-5718-1994-1242056-8, JSTOR 2153575