A BINARY TO RESIDUE CONVERSION USING NEW PROPOSED NON-COPRIME MODULI SET

Natarajan Meghanathan et al. (Eds) : NETCOM, NCS, WiMoNe, GRAPH-HOC, SPM, CSEIT - 2016
pp. 147– 159, 2016. © CS & IT-CSCP 2016 DOI : 10.5121/csit.2016.61513
A BINARY TO RESIDUE CONVERSION
USING NEW PROPOSED NON-COPRIME
MODULI SET
Mansour Bader1
, Andraws Swidan2
, Mazin Al-hadidi3
and Baha
Rababah4
1
Department of Computer Engineering, Jordan University, Amman, Jordan
mansoor259@yahoo.com
2
Department of Computer Engineering, Jordan University, Amman, Jordan
sweidan@ju.edu.jo
3
Department of Computer Engineering, Al-Balqa'a Applied University,
Salt, Jordan
trueamman@yahoo.com
4
Department of Computer Engineering, University of Portsmouth,
Portsmouth, UK
baharababah@yahoo.com
ABSTRACT
Residue Number System is generally supposed to use co-prime moduli set. Non-coprime moduli
sets are a field in RNS which is little studied. That's why this work was devoted to them. The
resources that discuss non-coprime in RNS are very limited. For the previous reasons, this
paper analyses the RNS conversion using suggested non-coprime moduli set.
This paper suggests a new non-coprime moduli set and investigates its performance. The
suggested new moduli set has the general representation as {2n
–2, 2n
, 2n
+2}, where n ∈
{2,3,…..,∞}. The calculations among the moduli are done with this n value. These moduli are 2
spaces apart on the numbers line from each other. This range helps in the algorithm’s
calculations as to be shown.
The proposed non-coprime moduli set is investigated. Conversion algorithm from Binary to
Residue is developed. Correctness of the algorithm was obtained through simulation program.
Conversion algorithm is implemented.
KEYWORDS
Forward Conversion, Residue Number System, Non-coprime Moduli Set
1. INTRODUCTION
Residue number system (RNS) is a subfield of finite field arithmetic [1]. It is widely used in
digital signal processing, image processing, FIR (Finite Impulse Response) filters, and IIR
(Infinite Impulse Response) filters because it is a carry-free system and high efficient in addition
and multiplication [2]. So, residue number system is used by most applications that need a high
degree of concurrency. A lot of researches in computer systems are enthusiastic to go through

148 Computer Science & Information Technology (CS & IT)
residue numbering system because of its characteristics such as, error detection and correction
(fault tolerant) [3], modularity, and embedded parallelism.
RNS allows dividing a large number into smaller sub numbers. Numbers are represented by
tipples which need less number of bits. The bits can be processed individually and in parallel
without carry between them. This improves computation time and simplifies hardware
implementation cost.
RNS has also the following advantages over conventional binary number system:
• Reducing the hardware complexity because the system is implemented by designing
smaller processing units.
• Improving the speed of operations since all of the tasks are performed in parallel.
• Efficient realization of the various building blocks needed such as adders, multipliers.
• The absence of carry propagation between the modulus channels makes the RNS
appealing for building parallel fast processors. This facilitates the realization of high-
speed, low-power arithmetic. This advantage is of paramount importance in embedded
processors, especially those found in portable devices, for which power consumption is
the most critical aspect of the design [4].
Because of these features, computer arithmeticians have historically promoted the RNS for high-
speed arithmetic-intensive applications [5].
The rest of this paper is organized as follows. In Section 2, overview of the new algorithm of
forward conversion is proposed. Section 3 presents the new non-coprime realization of the
proposed forward converter. The hardware implementation of the moduli set is presented in
section 4, while the paper is concluded in Section 5.
2. NEW ALGORITHM OF THE FORWARD CONVERSION OVERVIEW
In working with RNS the following three main terminologies are used:
1. Moduli set: defined in terms of relatively prime moduli where the ith
modulus presented
by m୧ and the gcd൫m୧, m୨൯ = 1, j ≠ i, i = 1,2, … , n. Numerous moduli sets can be
used. The characteristics of RNS based systems depend on the moduli set chosen.
2. Dynamic range (M): this is equal to the product of mi terms; M = ∏ m୧
୬
୧ୀଵ , and denotes
the interval of integers that can be represented uniquely in the RNS using the specific
moduli set.
3. Residues: to represent any number X in RNS we find x୧ = X mod m୧ for all m୧ moduli.
The number is represented as X = xଵ, xଶ, xଷ.
RNS based processing units are generally composed of: Forward convertor, arithmetic and logic
unit (ALU) and a reverse convertor shown in figure 1 [6].
Conversion from Binary to Residue is called forward conversion. This conversion is used in order
to process numbers in Residue format, because it is faster and it is easier for human being
understood. To use the Residue Number System efficiently one has to interface it with the real
world, the numbers should be converted from usual representation either binary or analog to
residue representation, this is the first step in using RNS. This step is a very complex and
demanding process, which acts as an academic challenge that restrict the use of RNS in many
practical applications. Many researches conducted to find the most efficient algorithm, hardware

Computer Science & Information Technology (CS & IT) 149
architectures and schemes either using special or arbitrary moduli sets in the implementation of
forward convertors in RNS.
In the forward translation of a binary number to its RNS equivalent, one of the most trivial,
classical expensive ways is to store all the residues and recall them based on the value of the
binary input.
Using the fact that the number can be represented as:
ܺ = ‫ݔ‬௡ିଵ ‫ݔ‬௡ିଶ … ‫ݔ‬ଵ‫ݔ‬଴ = ∑ ‫ݔ‬௝2௝௡ିଵ
௝ୀ଴ (1)
Figure 1. Block diagram of RNS based processing.
It is clear that this is a memory consuming process where you have to store all values in a lookup
table that typically consists of ROM, and for complex applications that require large number
representation, the size of memory will increase dramatically and thus increasing the cost.
The other implementation is to have special moduli set representation used in the conversion
process, consist of three, four, or five bits, the design of these convertors is based on using carry
save adder (CSA).
2.1. Non-coprime Moduli Sets Overview
Initially, only RNS with co-prime moduli set was investigated and used. Non-coprime had drawn
the attention of research only lately. Thus the knowledge of non-coprime characteristics has been
obtained from the co-prime one's as going to be seen through this chapter sections.

It is relatively easy to convert even numbers into their residue numbers representation. Numbers
as 2n
, like 16 which is pow(2,4) or in form that we are familiar with now 24
, and 14 which is 24
–
2 and 18 which is 24
+2, are not co-primed with each other since there is a common factor
between them which is number 2.
The usual way to compute a mod m is to take the remainder after integer division. This is straight
forward when the operands are within the range of the available divide hardware, but the divide
operation is known to be a slow arithmetic operation. Some small microcontrollers have no divide
hardware, and it is occasionally necessary to divide very large numbers outside the range that can
be done using the available hardware [7].
It can be faster to take the modulus directly than to use the divide instruction when the
modulus m is constant, even where there is a hardware divide instruction. Rules of divisibility
mentioned in table 1 become even more valuable on machines without a hardware divide
instruction or where the numbers involved are out of range.
Table 1. Some of divisibility check rules applied to decimal system.
2.2. Some Math Identities Review
2.2.1. Single General Rule
All divisibility check rules mentioned in table 1 are actually special cases of a single general rule.
Given that:
a is represented in number base b
a mod m = ( (b mod m)(a/b) + (a mod b) ) mod m
In the case of divisibility by 2, 5 and 10 for base 10, the term (b mod m) is zero because 2, 5 and
10 all divide evenly into 10. As a result, the divisibility test simplifies to asking whether
(a mod b), that is, the least significant digit of the number, is evenly divisible.
In the case of divisibility by 3 or 9 in base 10, the term (b mod m) is one. As a result, the
multiplier for the first term is one. Applying the formula recursively leads to the simple sum of
the digits [7].
2.2.2. The Trivial Case: Mod 2, Mod 4, Mod 2n
Computing modulus for powers of two is trivial on a binary computer, the term (b mod m) is
zero, so we just take the modulus by examining the least significant i bits of the binary
representation:
a mod 2i = a & (2i –1)
Thus, for a mod 2, we use a & 1, for a mod 4, we use a & 3, and for a mod 8, we use a & 7.
Divisibility by n Check
10 The least significant decimal digit is zero.
2 The least significant decimal digit is even.
5 The least significant decimal digit is 0 or 5
3 Sum of the decimal digits is divisible by 3.
9 Sum of the decimal digits is divisible by 9

Recall that the & operator means logical and. When applied to integers, this computes each bit of
the result as the and of the corresponding bits of the operands. For all nonzero positive integers i,
the binary representation of 2i
–1 consists of i consecutive one bits, so anding with 2i
–1 preserves
the least significant i bits of the operand while forcing all more significant bits to zero [7].
The problem is more interesting when the modulus is not a power of two only.
2.2.3. Mersenne’s Number: Mod 3, Mod 7, Mod 2n
-1
In mathematics, a Mersenne prime is a prime number that is one less than a power of two, i.e. 2n
-
1. That is, it is a prime number that can be written in the form Mn = 2n
− 1 for some integer n.
They are named after Marin Mersenne, a French Minim friar, who studied them in the early
17th
century [8].
Consider the problem of computing a mod 3 in binary number system. Note that 4 mod 3 is 1, so:
a mod 3 = ( (a/4) + (a mod 4) ) mod 3
That is, a mod 3 can be computed from the sum of the digits of the number in base 4. Base 4 is
convenient because each base 4 digit of the number consists of 2 bits of the binary represenation;
thus a mod 4 can be computed using a & 3 and a / 4 can be computed using a >> 2.
The number 3 is a Mersenne number, that is, one less than a power of two. The property noted
above is true of all Mersenne numbers. Thus, we can compute a mod 7 or a mod 15 on a binary
computer using:
a mod 7 = ( (a/8) + (a mod 8) ) mod 7
a mod 15 = ( (a/16) + (a mod 16) ) mod 15
Recall that a >> b shifts the binary representation of a left a total of b places. As with logical and,
this is a very inexpensive operation on a binary computer, and the effect is the same as
dividing a by 2b [7].
In this paper the problem is more interesting when the modulus is in a different shape of a power
of two, where it is in Mod (2n
– 2), Mod (2n
+ 2) consequently as a new moduli set proposed
along with Mod (2n
), that are going to be discussed in the next section.
Our work is done by suggesting the new moduli set {2n
– 2, 2n
, 2n
+ 2} and proposing new
conversion algorithms upon this new non-coprime moduli set. The next coming section will
discuss the background of non-coprime moduli sets.
2.3. Our New Non-coprime Moduli Set Overview
As its name shows "non-coprime" means the non-coprimality among its modulus numbers. Non-
coprime moduli sets can be used for error detection and correction purposes [9]. This non-
coprimality could be shown in theorems and by examples to prove them too.
Observation 1: 2k
− 2 is not relatively prime to 2k
, where k is a positive natural number.
It is obvious that 1 is not the only prime divisor of 2k
and 2k
– 2, since they are both even and 2k
–
2 is a multiple of 2 (i.e. a number multiplied by 2). Thus, there is a common divisor of them
rather than 1, which is number 2 in this case. So 2k
−2 is not relatively prime to 2k
.
Example 1:
Let us take K = 4, in this case the two numbers of the theorem one would be 16 and 14

consequently, and it is obvious that there is a common divisor which is 2 between them when we
try to bring them back to their elementary elements.
Observation 2: 2k
+ 2 is not relatively prime to 2k
, where k is a positive natural number.
It is obvious that 1 is not the only prime divisor of 2k
and 2k
+ 2 , since they are both even and 2k
+
2 is a multiple of 2 (i.e. a number multiplied by 2). Thus, there is a common divisor of them other
than 1, which is number 2 in this case. So 2k
+2 is not relatively prime to 2k
.
Example2:
Again let us take k = 4, in this case the two numbers of the theorem two would be 16 and 18
consequently, and it is obvious that there is a common divisor which is 2 between them when we
try to bring them back to their elementary elements.
2.4. Properties of Non-coprime Moduli Set
2.4.1 Dynamic Range of Our Non-coprime Moduli Set
This property of having a common divisor other than number 1 led to the non-coprime moduli set
when gathering the two theorems above. Based on these theorems the moduli set {2n - 2, 2n, 2n +
2} is non-coprime. The dynamic range of co-prime moduli set (M) is equal to M = ∏ m୧
୬
୧ୀଵ . In
the non-coprime case it is M = (∏ m୧
୬
୧ୀଵ )/4, it is 1/4 of the size of the co-primed one {2n – 1,
2n, 2n + 1}. This quarter comes from the multiplication of the common divisor (i.e. 2) between its
modulus numbers leading to number 4 which can not be multiplied to form the usual (M), thus its
size is less than the co-prime one. However it is important to know other characteristics such as
its uniqueness and bits representation.
Definition [9]: We define a non-prime moduli set as (ml, ..., mk), where gcd (mi,mj) = l may not
be satisfied for some i and j. The least common multiple of the moduli (m1, ..., mk), denoted as:
M = lcm(m1, ..., mk), is the dynamic range of (m1, ..., mk) . For any decimal number y∈[O, M –
1), y has a unique representation as (y1, ...,yk), where y = yi mod mi, 0 ≤ yi< mi.
If the residue number (y1, ...,yk) is consistent, the decimal number it represents can be found
using the CRT theorem, where M = lcm(m1, ..., mk) [9].
To find M through a formula as the one of the co-prime, we suggest the following formula due to
the special non-coprime moduli set:
M = (∏ m୧
୬
୧ୀଵ )/4 (2)
2.4.2 Uniqueness Verification
After dealing with the major and most important part of RNS in the previous section, it is
important to discuss the moduli set uniqueness when converted from decimal form (as we see it)
into RNS form (as to be implemented). In this section a mathematical proof is going to be
presented powered by table 2 to show the uniqueness itself.
As a mathematical proof, we already know that 6, 8 and 10 are not co-primed with each other, in
this case the LCM should be used as the previous section showed. Thus we have to return them
back into their fundamental factors, for 6 which equal 2*3, for 8 it equals 23 and for 10 it is 2*5,
now by taking the non common numbers without repetition and multiplying them we see that:

3*23*5 which equals 120, so M = 120. This is true for all other moduli sets chosen with the form
(2n-2 ,2n, 2n +2), and this is the non-coprime moduli set that we have got and found our research
on it for thesis level responsibilities, since the pre-moduli value (i.e. 2n -2) and the post moduli
one (i.e. 2n +2) are multiples of 2 and are un-coprimed with each other, but they are co-prime
with each other without it (as in 3 and 5 for the example above, when neglecting the
multiplication of them with number 2). This fact is also true for all other moduli sets used; this is
because they are 2 numbers relatively co-primed with each other and they are always odd.
The uniqueness interval was verified by simulation program, the results of a program using the
moduli set (6,8,10) was simulated, at which each iteration is repeated every 6*8*10/4 times,
which is in this case the number 120 (that is ‘M’ for the moduli set itself), and thus the
uniqueness is guaranteed under the non-coprime moduli set within its dynamic range.
3. FORWARD CONVERSION ALGORITHM
This section has three sub sections of it as the number of the modulus numbers forming the
moduli set are three.
3.1 Pre-modulus Algorithm Implementation
The binary representation led to this part's discovery, since its values are prime after dividing it by
2. We will give you how it works in words first then the algorithm of calculation
|X|2n
-2 can be represented in figure 1.
The idea of its binary cutting circles around the common factor which is number 2 in this case, as
the previous sections showed that after dividing the pre number by the common factor the result
is 2n-1
– 1. Since the pre value has a 2n
– 2 shape, then the cut of its binary representation would
be in (n cut at the beginning as the co-prime algorithm did, but for the rest part it is taken (n-1)
each time) starting from LSB again.
3.2 Middle-modulus Part Algorithm
No change is done on it, so we refer you to [15] – [17] where obtaining the residue of X with
respect to modulus 2n
is the easiest operation.
3.3 Post-modulus Algorithm Implementation
The binary representation led to its working algorithm, since its value is prime after dividing it by
2. Again we will give you how it works in words first then we will put them in the flowchart of
figure 2 to easily understand it.
The idea of its binary cutting circles around the common factor which is number 2 in this case, as
the pre-modulus section showed. Here dividing the post number by the common factor the result
is 2n
-1 + 1, and since the post value has a 2n
+ 2 shape, then the cut of its binary representation
would be in (n cut at the beginning as the co-prime algorithm did, but for the rest parts it is taken
(n-1) each time) starting from LSB again as the pre-modulus did, some examples will show how
this is done.

Figure 1. Computing the residue with respect to 2n
-2

Figure 2. Computing the residue with respect to 2n
+ 2
4. ALGORITH IMPLEMENTATION
In this section the implementation of the new non-coprime moduli set converter's implementation
is done through hardware block diagrams. As described in the previous section, the converter has

3 stages of computing the modulus; they are (pre-modulus '2n
– 2', middle modulus '2n
' and post-
modulus '2n
+ 2'). So this section also consists of three hardware implementation blocks of them.
4.1. Pre-modulus Hardware Implementation
From the proposed algorithm presented for the pre-modulus calculation discussed in the flowchart
of figure 1, we can come with this block diagram for the hardware design of it in figure 3.
Notice that the start for cutting is from LSB side to the MSB. The DR for any n value is
calculated as was shown in formula (2), thus any number inside the range of M has 3 part cuts -at
most-. For example let n = 5, M = 15*32*17 = 8160, so the numbers inside the DR are
{0,1,2,……,8159}. However 8159 is represented in binary form as (1111111011111) which
consist of 13 digits and it is equal to 5 + 4 + 4 as the cuts presented by its algorithm showed. This
is true for all n values, so we will name the first n-bits cut A, the second (n-1) bits cut B and the
final part of (n-1) bits is C.
3n-2 Input X
C B A
(n-1) n
1
(n-1)
n
1
n
Figure 3. Calculation of Pre-modulus (2n
– 2).
4.2 Middle-modulus Hardware Implementation
No change is done on it, so we refer you to [15]-[17]. Obtaining the residue of X with respect to
modulus 2n
is the easiest operation. Block diagram of it is shown in figure 4.
(n-1) bits (n-1) bits n-bits
Full Adder
Full Adder

3n-2 Input X
n
Figure 4. Calculation of Middle-modulus (2n).
4.3 Post-modulus Hardware Implementation
From the proposed algorithm presented for the post-modulus (2n
+ 2) calculation discussed in the
flowchart of figure 2, we can come with this block diagram for the hardware design of it in figure
5.
Notice that the start for cutting is from LSB side to the MSB. The DR for any n value is
calculated as was shown in formula (2), thus any number inside the range of M has 3 part cuts -at
most-. For example let n = 5, M = 15*32*17 = 8160, so the numbers inside the DR are
{0,1,2,……,8159}. However 8159 is represented in binary form as (1111111011111) which
consists of 13 digits and it is equal to 5 + 4 + 4 as the cuts presented by its algorithm showed.
This is true for all n values, so we will name the first n-bits cut A, the second (n-1) bits cut B and
the final part of (n-1) bits is C.
C B A
(n-1) bits n-bits
(n-1) bits 1
n-bits
1
n -bits
n-bits
Figure 5. Calculation of Post-modulus (2n + 2).
(n-1) bits (n-1) bits n-bits
Additive
inverse
X input of size (3n-2)
Full Adder
Full Adder

5. CONCLUSIONS
A new non-coprime moduli set has been proposed. A general formula for the dynamic range was
derived. Algorithm of the special non-coprime moduli set has been suggested. The uniqueness for
the new special non-coprime moduli set just as the co-prime one's among DR has been verified.
This research revealed that non-coprime moduli set may be suitable for wide variety of cases not
limited to co-prime only.
ACKNOWLEDGEMENTS
The authors would like to thank everyone.
REFERENCES
[1] Neha Singh, (2008), "An overview of Residue Number System". National Seminar on Devices,
Circuits & Communication.
[2] Chaves, R., and Sousa, L. (2007) "Improving residue number system multiplication with more
balanced moduli sets and enhanced modular arithmetic structures". Computers &Digital Techniques
IET, Volume 1, Issue 5, pages 472-480.
[3] Modiri, Samira, Movaghar, and Barati (2012) "Study of error control capability for the new moduli
set {22n+ 1+ 2n-1, 22n+ 1-1, 2n-1, 23n, 23n+ 1-1}". Journal of Advanced Computer Science &
Technology, Vol. 1, Pages: 176-186.
[4] Abdelfattah (2011) “Data Conversion in Residue Number System”. McGill University.
[5] Bajard and Plantard(2004) "RNS bases and conversions". Proceedings of Advanced Signal Processing
Algorithms, Architectures, and Implementations XIV SPIE, Vol. 5559,Page:61-75.
[6] William A. Chren, Jr., (2005), "Residue Number System Arithmetic Circuits With Built-in Self Test".
United States Patent 6886123 B1.
[7] http://homepage.cs.uiowa.edu /~jones/bcd/mod.shtml, last accessed on October, 2015.
[8] https://en.wikipedia.org/wiki/Mersenne_prime, last accessed on October, 2015.
[9] Y. Wang.(1998) "New Chinese Remainder Theorems". In proceedings of the Thirty Second Asilomar
Conference on Signals, Systems and Computers, Pages: 165-171.
[10] Vidhyalakshmi.M,Prof.Satyabama . (2014) “Design and Implementation of Efficient Binary to
Residue Converter Using Moduli Method”. International Journal of Innovative Research in Computer
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[11] Shende, Radha, and Zode.( 2012) "Efficient design 2k− 1 binary to residue converter." On
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[12] Omondi, Amos, and Premkuar.( 2007) “Residue number systems: theory and implementation”.
Imperial College Press.
[13] Mohan,and Ananda(2002) "Residue number systems: algorithms and architectures". Springer, vol.1,
pages:1-268.
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[15] Hiasat, and Sweidan(2003) "Residue number system to binary converter for the moduli set (2n− 1, 2n,
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AUTHORS
Mansour Bader holds a MSc in computer engineering and networks, University of
Jordan, Jordan, 2016. BSc Computer Engineering, Al-Balqa Applied University,
Jordan, 2008. He is a technical support engineer of computer networks at computer
center of Al-Balqa Applied University for 8 years and a half.
Dr. Andraws I. Swidan was born in Al-Karak Jordan in 1954. He received his
diploma in Computer Engineering (with honours) and Ph.D. in Computer Engineering
from LETI Ulianov Lenin, Sanct Peterburg (Leningrad), Russia in 1979 and 1982
respectively. He Joined the Electrical Engineering Department at the University of
Jordan in 1983 and was one of the founders of the Computer Engineering Department
at the University of Jordan in 1999. Since then he is a professor at the department. He
is also an Adjunct Professor with the ECE department of the McGill University,
Montreal, Canada. He holds several technical certifications among which the CISSP. He is an IEEE
member, Jordanian Engineering Association member Quebec College of engineers member. He is a Canada
Professional Engineer (The province of Quebec). He was member of several national and international
scientific committees. He holds several patents and tens of publications.
His main areas of research interest are: computer arithmetic, computer security, encryption algorithms.
Mazin Al-hadidi PhD. in Engineering Science (Computing Machines, Systems and
Networks), Institute of Problems of Simulation in Power Engineering Academy of
Science, Ukraine/Kiev .1990-1994, with grade Excellent.Bachelor and Master Degrees
in Engineering (Computer and intellectual systems and networks) Kiev Institute of Civil
Aviation Engineers, as a governmental scholarship, Soviet Union / Kiev, 1984-1990,
with grade very good.General Secondary 12 Years Certificate in the Science branch,
Jordan/Al-Salt, 1984, with grade very good.
Baha Rababah has a MSc of computer network administration and management (with
Distinction), University of Portsmouth, UK, 2015. BSc Computer Engineering, Al-
Balqa Applied University, Jordan, 2010. He is a lecturer of computer networks subjects
in Keys Training and Solution, Irbid, Jordan.

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