=Paper=
{{Paper
|id=Vol-2853/short36
|storemode=property
|title=Same Bit-Size Moduli Formation of Residue Number System for Application in Asymmetric Cryptography
|pdfUrl=https://ceur-ws.org/Vol-2853/short36.pdf
|volume=Vol-2853
|authors=Mykhailo Kasianchuk,Ihor Yakymenko,Vasyl Yatskiv,Stepan Ivasiev,Andriy Sverstiuk
|dblpUrl=https://dblp.org/rec/conf/intelitsis/KasianchukYYIS21
}}
==Same Bit-Size Moduli Formation of Residue Number System for Application in Asymmetric Cryptography==
<pdf width="1500px">https://ceur-ws.org/Vol-2853/short36.pdf</pdf>
<pre>
Same Bit-Size Moduli Formation of Residue Number System for
Application in Asymmetric Cryptography
Mykhailo Kasianchuka, Ihor Yakymenkoa, Vasyl Yatskiva, Stepan Ivasieva and Andriy
Sverstiukb
a
    West Ukrainian National University, 11 Lvivska str., Ternopil, 46009, Ukraine
b
    I. Gorbachevsky Ternopil National Medical University, 1 Maidan Voli, Ternopil, 46001,Ukraine


                 Abstract
                 This paper presents three methods (factorization, exhaustive search and replacement) of four
                 equal bit size moduli sets formation in a residue number system modified perfect form. This
                 allows using the bit grid registers more efficiently. Such problem is relevant for asymmetric
                 cryptography and noise-protected coding algorithms. The theoretical bases of residue number
                 system, its perfect and modified perfect forms are considered, their advantages and
                 disadvantages are defined. It is shown that the most commonly used moduli in the form of
                 power of two, Mersenne numbers and Fermat numbers require searching for the inverse
                 element and multiplying by it, which makes it difficult to recover a decimal number from its
                 residues using the Chinese remainder theorem. A modified perfect form of the residue
                 number system simplifies this procedure. The graphical dependency of the fourth modulo on
                 two prior ones with one known modulo is presented. Different bit sizes of moduli sets are
                 considered. It is shown that in sets of four modulo with the same bit size in a modified
                 perfect form of residue number system, the first and fourth moduli are negative, second and
                 third are positive.

                 Keywords 1
                 Residue number system, modified perfect form, computing range, modulo system, speed,
                 asymmetric cryptography.

1. Introduction
    Nowadays the rapid progress in the information technology area, in particular in mission-critical
applications, leads to the new demands for improved reliability, performance and productivity of
various computing systems [1]. Modern requirements for reliability of information transmission using
technical means lead to a decrease in productivity or increase in time and computational complexity
and, accordingly, to increased energy consumption [2]. On the other hand, economic factors and
development level of the technical means also impose appropriate restrictions. Therefore, it is advised
to use special code systems that do not have such restrictions to solve these problems [3]. However,
existing approaches and methods for data transmission and processing, that operate in positional
numeral system (PNS), can not achieve increased requirements data processing reliability without
reducing the performance of computing system with limited hardware and economic resources [4, 5].
It's caused by such disadvantages of PNS as high digit capacity, strictly consistent structure and the
presence of inter-bit transfers, that complicate the architecture of computing systems and reduce their
speed [6].


IntelITSIS’2021: 2nd International Workshop on Intelligent Information Technologies and Systems of Information Security, March 24–26,
2021, Khmelnytskyi, Ukraine
EMAIL: kasyanchuk@ukr.net (M. Kasianchuk); iyakymenko@ukr.net (I. Yakymenko); jazkiv@ukr.net (V. Yatskiv);
stepan.ivasiev@gmail.com (S. Ivasiev); sverstyuk@tdmu.edu.ua (A. Sverstiuk);
ORCID: 0000-0002-4469-8055 (M. Kasianchuk); 0000-0003-3446-1596 (I. Yakymenko); 0000-0001-9778-6625 (V. Yatskiv); 0000-0003-
2243-5956 (S. Ivasiev); 0000-0001-8644-0776 (A. Sverstiuk)
            © 2021 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)
   The most relevant task is the processing speed improvement for large amounts of numerical data in
asymmetric cryptography [7] and noise-protected coding problems during data transmission [8-10].
One of the possible ways of solving it is to use non-positional number systems, in particular, the
residue number system (RNS). Data processing and transmission in RNS has a number of advantages
due to independence, lack of inter-bit transfers, low bit size and equality of residues, as well as
possibility of parallel arithmetic operations execution. However, currently RNS is used only for
solving some specialized problems [11-16], due to required conversion of binary code, which is used
by universal computers and data processing devices, into RNS code, which allows to parallelize
computational processes [17, 18]. In addition, RNS has a number of disadvantages that have slowed
down its development, in particular, difficulties in performing division and numbers comparison
operations [19]. But since the main operations in asymmetric cryptography are multiplication and
exponentiation, the use of RNS becomes a very effective tool for processing multi-bit numbers [20] in
asymmetric encryption of information flows. Furthermore there are effective correction codes
developed for RNS, that are able to detect and correct error packages [21].

2. Theoretical basics of RNS and its usage in asymmetric cryptosystems

   Let us consider positive pairwise coprime integers р1 , р2 ,..., р z , which are called bases or system
                                              z
modulo (z-moduli count). Lets define Р = ∏ pi . This value represents the range of numbers in the
                                             i =1
selected moduli system. RNS is a non-positional number system in which the nonnegative integer N
can be presented as a set of nonnegative residues from division of this number on the chosen bases of
the system, such as
                                        bi=Nmod pi.                                                    (1)
   The RNS usage in computational algorithms is possible due to the presence of a certain
isomorphism between mathematical operations on integers and the corresponding operations on the
system of nonnegative integers over individual moduli. Moreover, the addition, multiplication and
exponentiation of any nonnegative integers are completely identical to the corresponding operations
performed on the residues system [18].
   The reverse conversion into a positional number system is usually based on the Chinese remainder
theorem [22]:
                                            z            
                                      N =  ∑ bi M i mi  mod P ,                                    (2)
                                            i =1         
              z                                                     P
where P = ∏ pi (inequality should be satisfied N<P), M i =             , to find mi multiplicative inverse
            i =1                                                    pi
element should be calculated:
                                              mi = M i−1 mod pi .                                    (3)
   Obviously, one of the ways to increase the speed of computers that use RNS is the choice of
specialized moduli sets, on which significantly depends the execution time of both modular and non-
modular operations. Therefore, for each specific application with the specified arithmetic operations,
hardware components and constraints, it is necessary to select the appropriate set of moduli. For
example, digital signal processing requires fewer modules than cryptography.
   In the vast majority of works moduli, such as 2k, 2k±1, are considered, which allows using of bit
grid registers effectively [23-25]. The worst modulo, which has the greatest execution complexity (it
can be the largest one or the complex type modulo), defines the general parameter of the direct
converter or the arithmetic channel.
   Additionally, RNS offers many different moduli sets of different types and different quantities for
certain applications, which significantly affect all parts of the hardware implementation, including
direct converters, modular arithmetic channels, inverse converters. In particular, [26-27] presents safe
and effective approaches for RNS usage on elliptic curves in cryptography. They are especially
effective as a response to attacks on the side channel of the source and for protection when the
malfunctions are injected in the computer system. In [28, 29], efficient algorithms for the RSA-
cryptographic system implementation based on RNS were developed, and the experimental studies
showed that they have greater speed and better resistance to brute force attacks compared to the
classical ones. In paper [30] the methods for fast execution of arithmetic operations such as addition,
multiplication and exponentiation in the modular number system with the implementation of
cryptographic transformations were developed, which show a significant reduction in the execution
time of the crypto algorithms basic operations.
    But all the considered approaches require calculation of the inverse element (3) and multiplying it
by (2), which significantly reduces the speed while recovering decimal number from the RNS [31-33].
This is, along with the difficulties in dividing and comparing numbers, the main drawback that has
slowed down the development of RNS. One of the options to simplify this procedure is to use
different forms of RNS. In particular, in [34] methods for perfect form (PF) of RNS formation, where
modules system рі is selected so that the values of all coefficients mi=1, are developed and
investigated. This approach eliminates the complicated procedure of finding a multiplicative inverse
element by modulus and multiplying by it while using a Chinese remainder theorem. However, the PF
of RNS has a limited application, because the bits of the corresponding moduli, and therefore the
residues, significantly differ, which leads to irrational usage of the bit grid registers. In addition, the
first moduli must be equal to 2 and 3, which limits its usage while building a modules system with the
same bit size.
    In [35] modified perfect form (MPF) of RNS was developed, which satisfies the condition:
                                M i mod pi = 1 , рі-1 or M i mod pi =±1.                                (4)
    In this case coefficients mi=1 and accordingly the sum in (2) change the sign. In addition to
                                                                                         z
eliminating mentioned drawback of the PF RNS, it significantly reduces the sum ∑ bi M i mi in (2),
                                                                                        i =1
which in many cases will not exceed the value of the calculation range. This will simplify the finding
of residue by modulo Р.
    However, currently there are no methods of constructing moduli systems that meet the conditions
of MPF RNS and have the same bit size, which will allow using the bit grid registers for efficient
asymmetric cryptography problems solving problems and noise-tolerant coding. This work is devoted
to the elimination of this shortcoming.

3. Methods for construction MPF RNS moduli system of the same bit size

3.1. Factorization method
  To construct a system of moduli with the same bit size for MPF RNS using the factorization
method, we write expression (4) as a system:
                                      M 1 mod p1 = ±1
                                      
                                      ...                                                        (5)
                                      M mod p = ±1.
                                       z         z
  Mathematical calculations similar to the ones performed in [33] lead to the following expression:
                                       z 1           1
                                      ∑      =γ ±      ,                                          (6)
                                                   z
                                     i =1 pi
                                                  ∏ pi
                                                   i =1
where γ =0, 1, 2, 3, ... .
    To simplify the calculations, the selected coefficient unlike the PF RNS, can be equal to 0, which
for a given number of moduli corresponds to the largest value Р. So the last equality can be described
as:
                               1     1     1            1     1              1
                                  +     +     + ... +       +    =±                         .       (7)
                               p1 p2 p3               p z −1 p z    p1 p2 p3 ... p z −1 p z
   Since the use of rational fractions necessarily presents rounding of the results with a certain
accuracy, for the convenience of software implementation of the developed method, expression (7)
should be reduced to a common denominator:
                                                               z
                                                              ∑ M i = ±1 .                                                           (8)
                                                             i =1
   Suppose that last two moduli pz-1 and pz are unknown. Therefore (8) can be defined as:
            p z −1 p z ( p2 p3... p z − 2 + p1 p3... p z − 2 + ... + p1 p2 ... p z −3 ) + p1 p2 ... p z − 2 ( p z −1 + p z ) = ±1.   (9)
   Let’s consider:
                                                       a, b − p1 p2 ... p z − 2
                                      p z −1, z =                                                  . (10)
                                    p2 p3 ... p z − 2 + p1 p3 ... p z − 2 + ... + p1 p2 ... p z −3
   After substituting (10) into (9) and performing some mathematical transformations we get a
condition that must be satisfied to determine a MPF RNS moduli set of any capacity with two
unknown moduli:
                      ± ( p2 p3 ... p z − 2 + p1 p3 ... p z − 2 + ... + p1 p2 ... p z −3 ) + ( p1 p2 p z − 2 )2 = ab .    (11)
   Therefore, the left part (10) should be factorized, and the parameters а and b are determined based
on it. Additionally, as a result of (10), the moduli рz and рz-1 should be integers, so
                     (a, b − p1 p2 ... p z −2 ) mod( p2 p3 ... p z −2 + p1 p3 ... p z −2 + ... + p1 p2 ... p z −3 ) = 0 . (12)
   Expressions (11) and (12) define the conditions to find any number of MPF RNS moduli, two of
which are unknown.

3.2. The method of exhaustive search
   For exhaustive search application it is advisable to choose formula (8). For four moduli we will
have following expression:
                                     p1 p2 p3 + p1 p2 p4 + p1 p3 p4 + p2 p3 p4 = ±1 .            (13)
   Figure 1 shows the graphical dependence of the modulo р4 on the values of the moduli р2 and р3.
As we can see the absolute value р4 increases with increasing of р2 and р3 at р1=-131. Integer values
of parameters р2, р3 and р4 create a moduli system that satisfies the conditions of MPF RNS.




                            p4




Figure 1: Graphical dependence of the modulo р4 on values of the moduli р2 and р3

3.3 Replacement method
  For four-moduli MPF RNS after replacement of p2, p3, p4=a, b, c-p1, and corresponding
mathematical transformations, formula (13) is converted to the following expression:
                                   abc − p12 (a + b + c ) = ±1 − 2 p12 .                                     (14)
   Reverse replacement allows returning to the original parameters:
                               ( p2 + p1 )( p3 + p1 )( p4 + p1 ) − p12 ( p2 + p3 + p4 + p1 ) = ±1 .          (15)
   Dividing the left and right parts by p12 , we get the conditions that must be satisfied to create MPF
RNS:
 ( p2 + p1 )( p3 + p1 )( p4 + p1 ) mod p12 = ±1, ( p2 + p1 )( p3 + p21 )( p4 + p1 ) ± 1 = p2 + p3 + p4 + p1 . (16)
                                                                   p1
   The first expression (16) indicates that ( p2 + p1 )( p3 + p1 )( p4 + p1 ) = k ⋅ p12 ± 1 , where k=1, 2, … .
   It means that k ⋅ p12 ± 1 must be decomposed into at least two factors (then the third will be 1). The
calculations show that in the vast majority of cases, following equality must be used to satisfy
conditions (16):
                                    ( p2 + p1 )( p3 + p1 )( p4 + p1 ) = − p12 + 1 .                          (17)

4. Results and discussion

4.1. Factorization method
   For example, let`s consider MPF RNS, which consists of four moduli. In this case conditions (10) -
(12) will be:
                     a, b − p1 p2
              p3,4 =
                       p1 + p2
                                                                     (            )
                                  ; ± ( p1 + p2 ) + ( p1 p2 )2 = ab; a, b − p1,2 mod( p1 + p2 ) = 0       (18)

   Suppose that the moduli bit size is 4 bits and р1=8, р2=-9. Then from (18) we get:
                                                                         5183 = 71 ⋅ 73
                          p3,4 = −(a, b + 72 ) and ab = ±1 + 5184 =                        .
                                                                         5185 = 5 ⋅17 ⋅ 61
   All possible options for systems with four moduli for MPF RNS with р1=8, р2=-9 and calculation
range are presented in table 1 (ab=5185) and table 2 (ab=5183).
Table 1
Possible options for systems with four moduli for MPF RNS with р1=8, р2=-9, ab=5185 and calculation
range
№               a                  b                       p3              p4                  P
1               1                  5185                    -73             -5257               27630792
2               -1                 5185                    -71             5113                26137656
3               5                  1037                    -77             -1109               6148296
4               -5                 -1037                   -67             965                 4655160
5               17                 305                     -89             -377                2415816
6               -17                -305                    -55             233                 922680
7               61                 85                      -133            -157                1503432
8               -61                -85                     -11             13                  10296
Table 2
Possible options for systems with four moduli for MPF RNS with р1=8, р2=-9, ab=5183 and calculation
range
  №                a                    b                       p3                 p4                 P
  1                1                  5183                     -73               -5255            27620280
  2               -1                 -5183                     -71               5111             26127432
  3               71                    73                    -143                -145             1492920
  4              -71                   -73                      -1                  1                72
   It is important to note that this table requires clarification of the moduli signs according to
expression (4). Therefore, a system of four 4-bit moduli with the same bit size, which satisfies the
MPF RNS conditions, will have following values: -8, 9, 11, -13.

4.2. The method of exhaustive search
   Table 3 presents possible options for sets of four 8-bit moduli for building MPF RNS.

Table 3
Possible options for sets of four 8-bit moduli for building MPF RNS
     №                   р1                       р2                   р3                   р4
     1                 -131                      134                  151                  -155
     2                 -134                      141                  143                  -151
     3                 -151                      178                  203                  -255
     4                 -169                      177                  179                  -188
     5                 -197                      199                  241                  -244
     6                 -208                      217                  219                  -229

    Obviously, this method is not suitable for finding a large number of high-bit size moduli, because
it is time consuming.

4.3 Replacement method
  Table 4 presents the procedure for finding moduli with the same bit size (5-7 bits) to form the
MPF RNS.

Table 4
 Procedure for finding moduli with the same bit size (5-7 bits) to form the MPF RNS.
 n, bit      р1     − p12 + 1 Factorization p2+p1        p3+p1       p4+p1     p2       p3          p4
     5     -19        -360          -2 ⋅3 ⋅5
                                      3 2          2          4       -45     21       23          -26
     6     -34       -1155         -3⋅5⋅7⋅11       3          5       -77     37       39          -43
     7     -76       -5775        -3⋅5 ⋅7⋅11
                                       2           5          7      -165     81       83          -89
     7    -103      -10608       -2 ⋅3⋅13⋅17
                                    4              6          8      -221     109      111        -118

   Based on results in tables 1-3 while creating the four-moduli MPF RNS, which are convenient for
application in asymmetric cryptosystems and noise-tolerant coding problems, for all moduli of the
same bit size the following relations are fulfilled: р1, р4 <0, р2, р3 >0.

5. Conclusions
    In this research the methods for the four-moduli sets of the same bit size creation in a modified
perfect form of residue number system are developed. This allows using bit grid registers more
efficiently. This problem is relevant for usage in asymmetric cryptography and noise-protected coding
algorithms. The theoretical bases of residue number system, its perfect and modified perfect forms are
considered, their advantages and disadvantages are defined. It is shown that the most commonly used
moduli are the power of two, Mersenne numbers and Fermat numbers require the finding the inverse
element and multiplying by it, which makes it difficult to recover a decimal number from its residuals
using the Chinese remainder theorem. A modified perfect form of residue number system simplifies
this procedure. Three methods of moduli system formation have been developed: factorization,
exhaustive search and replacement. The graphical dependence of the fourth modulo on two previous
ones with one known modulo is presented. Different bit sizes of moduli sets are considered. It is
shown that in four-moduli sets of the same bit size in a modified perfect form of residue number
system, the first and fourth moduli are negative, the second and third - positive.

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