=Paper=
{{Paper
|id=Vol-3137/paper15
|storemode=property
|title=Method for Generating Pseudorandom Sequence of Permutations Based on Linear Congruential Generator
|pdfUrl=https://ceur-ws.org/Vol-3137/paper15.pdf
|volume=Vol-3137
|authors=Emil Faure,Eugene Fedorov,Iryna Myronets,Svitlana Sysoienko
|dblpUrl=https://dblp.org/rec/conf/cmis/FaureFMS22
}}
==Method for Generating Pseudorandom Sequence of Permutations Based on Linear Congruential Generator==
https://ceur-ws.org/Vol-3137/paper15.pdf
Method for Generating Pseudorandom Sequence of
Permutations Based on Linear Congruential Generator
Emil Faure1,2, Eugene Fedorov1, Iryna Myronets1 and Svitlana Sysoienko1
1
Cherkasy State Technological University, Shevchenko Blvd., 460, Cherkasy, 18006, Ukraine
2
The State Scientific and Research Institute of Cybersecurity Technologies and Information Protection of the
State Service for Special Communications and Information Protection of Ukraine, M. Zaliznyaka Str., 3, bl. 6,
Kyiv, 03142, Ukraine
Abstract
The results of the study of the graph of states of a linear congruential generator (LCG) are
considered and theoretically substantiated. A model of a generalized graph of LCG states has
been developed. It represents each connected component of the graph in the form of cycles
equipped with tree products, allows classifying the types of connectivity components of the
graph of LCG states and investigating the influence of parameters on its topology. A method
for generating a pseudorandom sequence (PRS) of numbers based on the linear congruential
method is presented. This method allows generating uniformly distributed numbers
regardless of the topology of the graph of LCG states and, consequently, minimizing the time
spent on choosing its parameters, and increasing the size of the space of their allowable
values to achieve the maximum period. Computer implementation of the algorithm for
generating PRS of permutations based on LCG with any type of graph of its states has
allowed increasing the speed of the generator compared to the permutation generator using
the modern Fisher-Yates algorithm.
Keywords 1
Pseudorandom sequence, permutation, shuffle, random interleaver, linear congruential
generator, graph of states, monad, topology, monad graph
1. Introduction
Pseudorandom sequences (PRSs) [1-2] are widely used to solve a wide class of problems. In
particular, they are used to protect information from unauthorized access [3-5], to control the integrity
of information [6], to form signals that provide covert data transmission and implement the modeling
of complex systems and objects [7-9], to form permutations for factorial coding of information [10-
12].
The linear congruential generator, the linear-feedback shift register (LFSR), and the additive
generator are the simplest ones in design, high performance and ones of the most commonly used
generators. Their design is often the basis of high-quality pseudorandom number generators (PRNGs)
with a long repetition period.
Linear congruential generator (LCG) is proposed by D. H. Lehmer in 1949 [13]. It implements a
recurrent relation ( f : S S transition function) of the form:
si K si 1 C , M
(1)
where K is the multiplier;
C is the growth;
M is the module, K , C , s0 Z M .
CMIS-2022: The Fifth International Workshop on Computer Modeling and Intelligent Systems, Zaporizhzhia, Ukraine, May 12, 2022
EMAIL: e.faure@chdtu.edu.ua (E. Faure); fedorovee75@ukr.net (E. Fedorov); i.myronets@chdtu.edu.ua (I. Myronets);
s.sysoienko@chdtu.edu.ua (S. Sysoienko)
ORCID: 0000-0002-2046-481X (E. Faure); 0000-0003-3841-7373 (E. Fedorov); 0000-0003-2007-9943 (I. Myronets); 0000-0002-0009-
337X (S. Sysoienko)
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
Obviously, si 0; M 1 , and the power of the space of LCG states S M .
Congruential sequence always forms repeating cycles [1].
LCGs are very sensitive to changes in parameters. Numerous works on the theory and application
of congruential generators are aimed at choosing their parameters and assessing the quality of the
obtained PRSs. Among them are both classical [1, 14-17] and modern works [18-22].
At the same time, despite the large number of studies on the choice of LCG parameters and
experimental evaluation of its properties, most of them are aimed at improving the "randomness" of
the formed sequence and do not take into account the structure of space of LCG states S .
Among the works devoted to the analysis of the structure of the space of LCG states, one of the
main ones is the thorough scientific work of G. Marsaglia [23]. In addition, the work [24] is devoted
to the development of the theory of PRS construction based on the LCG and LFSR.
The analysis of the simplest PRNGs, LCG and LFSR, shows their limitations due to the need to
select parameters in order to ensure necessary PRS statistical properties. In particular, the allowable
values of the generator parameters to ensure the maximum PRS period are shown in Table 1.
Table 1
PRNG parameters to achieve the maximum PRS period
Valid values of PRNG Size of space of permissible
Method Period
parameters values of PRNG parameters
1) Gcd С , М 1 ;
2) K 1 p 0 for simple M P ,
Linear congruential where P is the number of
T M p: M p 0;
method [13] K values that satisfy
3) if M 4 0 K 1 4 0 conditions 2 and 3
Corresponds to the number
Method based on LFSR Generator polynomial Gn x
[25‐27] T 2 1
n
of primitive polynomials
is primitive one of n degree
Method for PRS
generating based on
concatenation of LCG T M Gcd K , M 1 M M
cycles [24]
Corresponds to the number
Method for PRS
Generator polynomial Gn x of polynomials of n degree
generating based on
concatenation of LFSR T 2n generates a cyclic structure that generate the cyclic
of the graph of LFSR states structure of the graph of
cycles [24]
LFSR states
The purpose of the work is to generate PRS of permutations with high performance and necessary
statistical properties without the need to select LCG parameters.
For the further study and analysis of PRNG construction based on sequential traversal of all
vertices of the graph of LCG states, it is necessary to perform an in-depth study of the structure of the
graph of LCG states, extended analysis of the influence of LCG parameters on the structure of its
graph, and, consequently, to develop the method for generating PRS based on linear congruential
method by sequentially traversing the contour of the graph of states of the generator.
2. Review of the literature
To study and generalize the structure of the graph of LCG states, we shall explore the main
approaches to forming graphs of states of PRS generating devices.
2.1. Cycle graphs
A cycle graph [28], also known as a simply n -cycle [29], is a graph containing n nodes and
consisting of a single cycle that passes through all its nodes. The cycle graph is denoted by Cn . The
number of vertices in Cn is equal to the number of edges, each vertex has a power of 2, any vertex is
incidental to two edges.
Cycle graphs are used, for example, to illustrate the structure of multiplicative groups M n
(Figure 1). Such graphs are formed by creating numbered nodes, one for each element of the
surplus class, and constructing cycles obtained by calculating i for i 1,2, . Each edge of such a
graph has a bidirectional character [28].
Figure 1: Graphs for some small‐order multiplicative groups (from [30])
In all graphs of Figure 1 the node with 1 is highlighted because it is a zero cycle:
j j 1 for any j . Next, we shall use the same notation to represent zero cycles in the
M
graph of LCG states.
Note that in Figure 1 not all represented graphs are cycle graphs. This follows from the fact that
multiplicative group by M module can be isomorphic to the product of several cyclic groups (for
example, M 8 C2 C2 , and M 15 C2 C4 ). In this case, the graph is a combination of several
cycle graphs.
To visually represent the structure of the graph of LCG states, it is necessary to use an oriented
cycle graph, an oriented version of the cycle graph, in which all arcs are directed in the same
direction.
2.2. Algebra of monads and topology of monad graphs
This paragraph, as well as its name, is based on the V. I. Arnold works [31-32].
According to [31], a monad is a representation of a finite set in itself. The monad graph has all
elements of this finite set as vertices, and oriented edges connect each element with its image.
In other words, a monad graph is an arbitrary finite oriented graph, from each vertex of which
exactly one edge emerges. Iterations of the monad lead any vertex to a cycle attractor.
According to [31], each connected component of the monad graph is a forest of root-oriented root
trees, the roots of which are connected by an oriented cycle (topologically circular) from the edges
connecting the tree roots.
In other words, connected components of any monad are cycle attractors, which are equipped with
root trees attached by their roots to each vertex of the cycle attractor [32]. The number of vertices of
the cycle can be equal to 1. In this case, the whole component is one root tree.
Each connectivity component of the graph of any representation of a finite set contains one and
only one cycle.
Let S be a finite group, and f : S S representation transforms each of its elements s S
according to the expression: f s K s C M , where K is the multiplier; C is the growth; M is
the module, K , C , s0 M .
In this paper, f : S S representation will be called the monad of S group.
According to [31], the symbols On , An , Tm , En will denote the following oriented graphs:
On = oriented cycle of n vertices;
An = connected graph of 2n vertices, which is a cycle of n length, equipped with n single-
edge trees, which are included one in each of n vertices;
T2n = root tree with 2n vertices and n floors except the root, which branches binarially on
1,, n 1 floors; the root is considered to be the zero floor, and it also includes two edges: one is
from itself and one is from a single vertex of the first floor;
En = root tree with n vertices, from each of which the edge leads directly to the root (so that
E2 A1 T2 );
Dn = 4n -vertex graph, consisting of On cycle of n length, equipped in each of its vertices
with three input edges (form together with this corresponding to the cycle vertex the root tree
D1 E4 ).
For example, graphs of monads for additive cyclic groups in the field n have the form shown in
Figure 2.
Figure 2: Graphs of monads for some simple cyclic additive groups (from [31])
According to [31], a monad that acts on the direct product X Y component by component:
A B x, y Ax By is called the A B product of A and B monads that act on X and Y ,
respectively. The number of elements of a monad product is equal to the product of the number of
elements of monad coefficients.
In [31] it is also shown that the graph of the monad product is the product of graph coefficients:
graph A B graph A graph B / An A1 On , Dn D1 On .
Multiplying any root tree T by On equips n -cycle On with root trees of T type with roots at all
points in the cycle.
3. Materials and methods
In this section, we shall investigate the LCG topology and generalize the graph of its states to
develop a method for generating permutation sequences.
3.1. Graphs of linear congruential generator
Examples of graphs of monads of S group for some LCG parameters are shown in Table 2.
Table 2
Oriented graphs of LCG states for some of its parameters
LCG parameters LCG parameters
Graph of states Graph of states
M K C M K C
3 1 5 4
0 1
6 4 3 7 2 3
0 4 2 2 3 6 5
0 2 3
5
0 1
7 6 1 1 6 5 7 4 6
2 6 4 3
4
0 1 3 5 7
4 2
6
7 3 2 6 8 4 2
3 1
5 4 2 0
0 4 5
3 1
7 6
8 5 1 2 6 8 7 3 0 1
5 7
4 3 2
0 3 4
2 0
8 3 1 3 7 5 1 8 7 2 2 7 6
6 4
1 5
0
7 5
1
6 0 2
8 1 6 3 7 2 6 8 6 4
3 1
5 4 4
Let us analyze the structures of graphs of monads of LCG S group. We shall summarize the
analyzed structures and present in Table 3 some graphs typical to LCG.
Extending the regularities given in [24], we give some graphs of LCG states and parameters for
which these graphs are typical:
1. OM – for:
a) M 2 , K 1 , C 1 ;
b) M 2 p , K 4l 1 , C 2m 1 , l , m 0 ;
c) M is simple, K 1 , C 1 ;
Table 3
Generalized graphs of LCG states
1. Generalized graph of cycles
Groups of cycles of ti length
i 1, ti 1, di ti M
i d Oi
i ti
2. Generalized graph of trees
Group of a n ‐ vertex trees with n
floors
a 2 , da n M dTan
3. Combinations of cycles and trees
A group of cycles of t length,
equipped in each of its vertices
with input edges n 1 , and a d En Ot kEn
group of root trees with n
vertices, from each of which the d T O kT
n1 t n1
edge leads directly to the root
n dt k M
A group of zero cycles with single‐
edge trees included in them,
equipped in each of their vertices d En A1
with root trees with n vertices,
from each of which the edge d T T
n1 21
leads directly to the root
dn M 2
A group of cycles of t length,
equipped with t a n ‐vertex trees
with n floors, and a n group of
a n ‐ vertex trees with n floors
n 2, a n dt k M
d Tan Ot kTan
2. dOt – for M 2 p , K 4l 1 , l 1 , and C 2m 1 , m 0 . Under these conditions:
a) for l 2 k 2 K 2k 1 , k 2 , t 2M K 1 , and d K 1 2 (or t 2 p l 1 , d 2l 1 )
take place;
b) for l 2k 1 K 8k 5 , k 1 , t M 2 , and d 2 take place;
c) for l 2k , k 1 , K 2r 1 , r 3 (that is k 2r 3 : k 2i 4 2 j 1 , and l 2i 3 2 j 1
K 2 2 j 1 1 for j 1;2
i 1 p i
1 , i 4; p 1 ), t M 2i 2 , d 2i 2 take place;
3. dOt O1 – for simple M , K 2 , C ;
4. a tree with a root, zero cycle, for М 2 p , K 2l : l ,0 l 2 p 1 , C 0,1,, 2 p 1 ,
p
3.2. Generalized graph of states of linear congruential generator
Analysis of possible graphs of LCG states shows that they can all be reduced to a single
configuration containing a set of d cycles of the same or different length, including zero cycles, and
a set of d ' precycles (trees) leading to cycles. Under these conditions
d d'
M ti t 'j , (2)
i 1 j 1
where ti is the length of the i th cycle;
t 'j is the number of vertices in the j th tree except for the root.
The generalized LCG graph can be represented as follows:
GLCG VLCG , ALCG ,
where VLCG is the set of vertices of the graph;
ALCG is the set of arcs of the graph.
In its turn,
VLCG vk vl' ,
d
where vk is the set of vertices belonging to the cycles, k 1, 2,, ti ;
i 1
d'
v is the set of vertices belonging to the trees, except for their roots, l 1, 2,, t ;
'
l
'
j
j 1
ALCG ak a ,
'
l
d
where ak is the set of arcs belonging to the cycles, k 1, 2,, ti ;
i 1
d'
a is the set of arcs belonging to the trees, l 1, 2,, t .
'
l
'
j
j 1
According to [33], for a simple M , the expressions d ' 0 , ti t for i 1, d 1 and t d 1 are
d 1
fair, and expression (2) takes the form: M t 1 .
i 1
We present a generalized graph of LCG states in terms of the theory of algebra of monads and the
topology of monad graphs.
The analysis of typical oriented graphs of LCG states shows that no more complex patterns, except
for the products of trees and cycles, are found in the graphs of monads of f : S S representation.
LCG graph is an inconnected combination of cycles equipped with tree products. Since En Tn1 O1 ,
At T21 Ot , Dt T41 Ot , each connected component of LCG graph can be represented as
Tan Tbm Ot . For example,
Ot Ta0 Tb0 Ot , and
At Ta0 T21 Ot T21 Tb0 Ot ,
En Ta0 Tn1 O1 Tn1 Tb0 O1 .
Then the generalized graph of LCG states has the form:
,
d
GLCG d i Ta ni Tb mi Oti (3)
i i
i 1
where d is the number of different types of connectivity components of the graph of LCG states;
d i is the number of connectivity components of the graph of LCG states of the B th type;
ai , ni , bi , mi , ti are parameters of connectivity components of the graph of LCG states of
th
the i type.
d
In this case di aini bimi ti M .
i 1
d
Determining the rules for calculating the number of graph components d i , their types d and
i 1
values of numbers ai , ni , bi , mi , ti through LCG parameters is beyond the scope of this work and
requires further research.
3.3. Method for generating sequences of permutations based on linear
congruential method
The method for generating LCG-based PRS is as follows.
1. If necessary (for example, to increase the speed of PRS generating or meet the requirements
for the spatial complexity of the algorithm that implements the proposed method), the type of
graph of LCG states, as well as the conditions to be met by K , C and M LCG parameters to
obtain a given type of structure are determined. Determining the type of the graph of LCG states
can be performed in accordance with typical graphs presented in Table 3. M parameter
determines the area of determination of pseudorandom variable. If the choice of the type of the
graph of states is not made, LCG parameters are determined arbitrarily, taking into account the
restrictions imposed on them.
2. If necessary (for example, for non-consecutive cycles of the graph of LCG states without
precycles (trees) to increase the speed of PRS generating), the representatives of each cycle of the
generator (boot vectors (BVs)) are determined and stored in memory.
3. The current LCG BV is determined by (random or deterministic) choosing from the set of
stored BVs, if this set is specified, or from the set of integers in the range 0, M 1 .
4. A PRS is formed by the LCG with given parameters until the generator forms unique
numbers. In the case of reappearance of any element (not necessarily equal to BV (for a graph
containing continuous cycles without precycles (trees), equal to BV)) the generation of the current
segment of the sequence is stopped.
5. A new current BV of LCG is determined by its (random or deterministic) choosing:
from the set of still unused BVs, if this set is specified;
from the set of integers of the range 0, M 1 except for the numbers present in the formed
part of the PRS.
6. PRS is formed for a given BV until the reappearance of the element in the formed sequence
(for a graph containing continuous cycles without precycles (trees), equal to the current BV).
7. Transition to item five until the shuffle of all BVs.
8. Transition to item three until the shuffle of all combinations consistently used in items three
and five of BV (if they are set and stored in memory).
9. Transition to item two until the shuffle of all BV combinations (if they are set and stored in
memory).
Thus, the proposed method allows performing concatenation not only of separate and disjoint
cycles in LCG graph, but also of precycles (trees), if they are contained therein.
In addition, the proposed approaches can be used to form PRS based on LFSR with an arbitrary
generator polynomial. This is because the use of a reducible polynomial as a generator one for LFSR
leads to an increase in the number and the change in the structure of connected components in the
generator graph of states.
4. Experiments and results
The proposed approaches to constructing devices for PRS generating based on LCG are used to
create software implementations of generators.
The size of the space of allowable values of LCG parameters for the above method is equal to
M 2 . The comparison with the corresponding indicators of the analogues in table 1 shows that this
method allows increasing the size of the space of allowable values of LCG parameters to achieve the
PRS period T M in M M times.
Let us study the speed of software implementation of the permutation generator based on the
developed method and compare it with the speed of the generator that implements the modern Fisher-
Yates algorithm [34]. For the sake of objectivity, the generators were implemented on one platform
and tested on one computer with fixed performance indicators. The results are presented in Figure 3.
Figure 3: Graphs of dependence of speed of permutation generators on M value
The speed of the developed generator exceeds the speed of the permutation generator using the
Fisher-Yates algorithm for M 125 , which expands the results obtained in [35].
It should be noted that PRS formed according to the proposed method is not cryptographically
stable and can not be used in "pure" form in cryptographic transformations, for example, as a gamma
for stream ciphers. However, the proposed approaches to PRS generating can be used to implement a
multi-stage encryption procedure.
5. Conclusions
The scientific novelty of the study is as follows. It is developed the model for generalized graph of
states of linear congruential generator, which allows to carry out the classification of types of
connectivity components of the graph of its states and to investigate the influence of parameters on its
topology. The developed model has allowed to improve the method for PRS generating based on
linear congruential method, which allows to form PRS of uniformly distributed numbers regardless of
the topology of the graph of states of linear congruential generator and, as a result, to minimize the
time spent on choosing its parameters and increase the size of the space of their allowable values to
achieve the PRS period T M in M М times.
Implementation of the algorithm for generating PRS of permutations based on LCG with any type
of the graph of its states has allowed to increase the speed of the generator compared to the
permutation generator based on PRNG LFIB78 using the Fisher-Yates algorithm for the permutation
order M 125 : in particular, for M 20 – in 2.1 times; M 50 – 1.6 times; M 100 – 1.2 times.
6. Acknowledgements
This research was funded by the Ministry of Education and Science of Ukraine, grant
number 0120U102607.
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