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 ${\displaystyle m=p_{1},\dots p_{r}}$ with arbitrary distinct primes ${\displaystyle p_{1},\dots ,p_{r}\geq 5}$ will be present here.

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

${\displaystyle y_{0}={\rm {seed}}}$

${\displaystyle y_{n+1}\equiv ay_{n}^{\varphi (m)-1}+b{\pmod {m}}{\text{, }}n\geqslant 0}$

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

Example[]

Let take m = 15 = ${\displaystyle 3\times 5\,a=2,b=3}$ 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

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

Theorem 1[]

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.

Discrepancy bounds of the GIC Generator[]

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.

See also[]

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

References[]

Notes[]

• Eichenauer-Herrmann, Jürgen (1994), On Generalized Inversive Congruential Pseudorandom Numbers (first ed.), American Mathematical Society, JSTOR 2153575