Secured Paillier Homomorphic Encryption Scheme Based on the Residue Number System

International Journal on Cryptography and Information Security (IJCIS), Vol. 12, No.1, March 2022
DOI:10.5121/ijcis.2022.12101 1
SECURED PAILLIER HOMOMORPHIC ENCRYPTION
SCHEME BASED ON THE RESIDUE NUMBER SYSTEM
Daniel Asiedu1
and Abdul-MuminSalifu2
1
Department of Computer Science, Tamale Technical University, Box 3 E/R,
Tamale, Ghana
2
Department of Computer Science, C. K. T. University of Technology and Applied
Sciences, Navrongo, Ghana
ABSTRACT
In this paper, we present an improved Paillier Cryptosystem for a secured data transmission based on the
Residue Number System (RNS). The current state of Paillier Cryptosystem allows the computation of the
plaintext from the cipher text without solving its security assumption of Decisional Composite Residuosity
or the knowledge of its private keys under mathematical attacks. The proposed RNS based cryptosystem
involving two stages of encryption and two stages of decryption has never been adequately studied before.
This paper attempts to solve by introducing two stages of encryption and two stages of decryption. The first
stage of the encryption process maintains the traditional Paillier encryption process and the second stage
process is the encryption using the recommended moduli set – by the RNS
Forward converter. At the first stage of the decryption process, our proposed RNS based reverse converter
is adopted and finally, the traditional Paillier decryption process will be used at the second stage of the
decryption process. Because the entire encryption technique is randomized, it can withstand chosen brute-
force attacks. The suggested algorithm's security study reveals that it has a wide key space
( , a high level resistance to key sensitivity attacks, and an acceptable level of resilience. In
terms of security, it has been discovered that the proposed system outperforms the present algorithm.
KEYWORDS
Cryptography, RNS, Information Security, Forward Converter, Reverse converter, Paillier Cryptosystem,
Dynamic Range.
1. INTRODUCTION
Cryptography is a branch of information science that explores methods for establishing secure
communication and using codes to protect plain text messages. It's when the original sender
sends a message or information to the intended receiver while preventing an adversary valid
access to cause any repetition. Cryptography's core idea is to allow two parties to communicate
via an unsecured channel in such a way that an adversary cannot decipher what is being
transmitted. Information security, often called Cryptography, is a major issue in data
communication networks. This is because, the broadcast signal may travel beyond the conversing
parties in both wired and wireless communications. With the correct equipment, anyone might
easily intercept the data being transferred. To prevent intruders from deciphering intercepted
signals, it is critical to encrypt data before transmission. Information security is extremely very
important and a serious consider decisive the standard of service in information transmission.
There's no such factor as excellent security; we'd like to concentrate additional on creating our
information troublesome to steal and making that meaning out of it.

2
Constructing and analyzing cryptographic protocols which are often trounced by adversaries and
in information security is often referred to as data integrity, data confidentiality, data privacy,
non-repudiation, reliability, and data authentication. In Cryptology, data encryption is classified
as Symmetric key cryptographic, Asymmetric key cryptographic, and Hash function. In
symmetric key cryptography, both the sender and receiver make use of a single key for
encryption and decryption. Commonly used ones are Block Cipher, DES (Data Encryption
System), Blowfish, RC2, and Stream Cipher. Asymmetric key cryptographic uses a couple of
keys for encryption and decryption process both for the sender and the receiver. Commonly used
ones are RSA, DSA, PKCs, Pailliar, and Elliptic curve. Hash function instead of using
predetermined keys uses mathematical equations by taking numerical data as input and produces
hash message as the resulting output. Commonly used ones are MD5, RIPEMD, Whirlpool, and
SHA. ATM cards, web encryption (HTTPS), computer passwords, time stamping, digital
signature, and electronic commerce are some of the areas of application of cryptography.
The Paillier cryptosystem is the most generally used public-key encryption system to hide
information from unauthorized access and different malicious activities due to its intensive
application in e-voting, e-cash and e-commerce systems. However, in (Asiedu & Salifu, 2020),
Asiedu and Salifu conducted a security risk assessment on Paillier Cryprosystem to identify
threats and weaknesses. It was found that, the Paillier Cryptosystem can be broken under a series
of mathematical attacks. That is, revealing the plaintext from the cipher text without solving its
security assumption of Decisional Composite Residuosity (DCRA) or the knowledge of its
private keys. As a result, the most valuable research is on improved Paillier public-key
cryptosystems.
The Residue Number System (RNS) is an integer number system that exhibits supporting
capabilities of carry-free addition, parallel computation, borrow-free subtraction, one step
multiplication without considering partial product which are the difficulties to binary and decimal
number system.
In this paper, the Residue Number System (RNS) is utilized to improve the Paillier public-key
cryptosystem by passing the cypher text from the traditional Paillier encryption scheme through a
smaller moduli set. Because the chosen moduli set is part of the private key, the key length is also
increased. Also, the intractability of solving its security assumption of Decisional Composite
Residuosity (DCRA) will not be used exclusively in this cryptosystem. In terms of security, the
suggested system outperforms the existing system.
2. OVERVIEW OF RNS
The Residue Number System (RNS) is an integer number system that supports parallel, carry-free
addition, borrow-free subtraction, and single-step multiplication with no partial product.
Therefore RNS offers the properties of parallelism (Flores, 1969). The inherent properties of
RNS have led to its intensive and widespread applications, such as image processing,
communications, Digital Signal Processing (DSP), Fast Fourier Transform (FFT), Digital
filtering, Discrete Cosine Transform (DCT), correlation, convolution, highly computing
applications, and cryptography (Schoinianakis, 2020).
Nonetheless, magnitude comparison, sign detection, moduli selection, overflow detection and
correction, data conversion, division, and other complex computing operations are still research
problems in RNS.

3
Forward conversion is the process of converting a conventional number system to a residue
number system, and Reverse/Backward conversion is the process of converting a residue number
system to a conventional number system, both of which are accomplished using the Chinese
Remainder Theorem or Mixed Radix Conversion or any variations of the two can be utilized to
achieve reverse conversion.
2.1. The Algebra of RNS
RNS is defined by a set of moduli set that are relatively prime to each other and
GCD where . The dynamic range denotes total permissible
numbers that can be represented by this RNS system. An integer X can be represented by the
residues where
2.2. Residue Representation
Given any base, the RNS representation, where are integers defined by a set of N
equations . Where, and is an integer so chosen that .
It is clear that is an integer value of a quotient which is denoted by . The quantity
is the least positive integer (remainder) of the division of by and is represented as
. can be rewritten as .
Example: Given that determine and the RNS representation
of if .
Solution: . .
. .
. Therefore the RNS representation of 25 is .
3. RELATED PREVIOUS WORKS
In 1999, a new probabilistic public cryptographic encryption method with a homomorphic
property was proposed by Pascal Paillier (Fontaine & Galand, 2007; Paillier, 1999). The Paillier
scheme is viewed as an extension of Okamoto-Uchiyama. The security assumption of the scheme
has been proven under Decisional Composite Residuosity Assumption (DCRA). With its additive
homomorphic property, the Paillier scheme has gained a lot of attention in numerous
applications, such as electronic voting, machine learning on encrypted data, threshold schemes,
and cloud computing (Albugmi et al., 2016; Shihab Ahmed & Zolkipli, 2016). The scheme is
based on computation over , n being RSA modulus. The security of the Paillier scheme is
based on the assumption that deciding nth composite residuosity: is considered
to be computationally difficult. That is, it is hard to determine whether is n-residue modulo
given as a composite number and as an integer.
Damgard, et al. (I. Damgård et al., 2010), in their paper titled “A Generalization of Paillier’s
Public-Key System with Applications to Electronic Voting”, proposed a useful application of
Paillier’s scheme in the area of Electronic Voting. Jurik (Jurik, 2003), in his thesis titled
“Paillier’s original scheme. Extensions to the Paillier Cryptosystem with Applications to
Cryptological Protocols”, proposed some useful length flexibility. This is based on the ability to
extend the plaintext space at encryption time rather than at key generation time, when the public

4
key is chosen, which was only available for symmetric ciphers in literature. In (Gupta & Sharma,
2013; Gupta Iti Sharma Asso, 2013), an asymmetric key encryption scheme with fully
homomorphic evaluation capabilities was proposed. The operations are matrix-based, that is, the
scheme consists of mapping the operations on integers to operations on a matrix. They further
include a protocol that uses the proposed scheme for private data processing in clouds. (Catalano
et al., 2001), this paper evaluates the hardcore bits of Paillier’s new trapdoor scheme. The
assumption was to prove that the least significant bit of , is a hard-core bit if we
assume computing residuosity classes is hard. In other words, we show that given a random
, if one can guess better than at random, then one can compute the whole
efficiently.
The related works so far focused on the following areas of the scheme under consideration,
Paillier Cryptosystem: Applications of the scheme (Jiang & Pang, 2020; Pettersen & Gjøsteen,
2016), Implementation of the scheme (Moore et al., 2014), Length Flexibility (I. B. Damgård et
al., 2003) and Cloud Computing (El Makkaoui et al., 2020; Moulay et al., 2017; Papisetty, 2017).
However, much attention has not been drawn to the stability of the Paillier Cryptosystem. In
other words, how can the scheme be broken without solving its security assumption of Decisional
Composite Residuosity (DCRA) or using its private key parameters until Asiedu and Salifu
(Asiedu & Salifu, 2020) proved that, the Paillier Cryptosystem can be broken under a series of
mathematical attacks without solving its security assumption of Decisional Composite
Residuosity (DCRA) or using its private key parameters.
This paper proposed a secured Paillier Cryptosystem using the inherent advantages of RNS to
overcome those security challenges, making the scheme secure and robust for full utilization in
the community of cryptography.
4. THE PROPOSED CRYPTOSYSTEM
Our proposed RNS based cryptosystem involves two stages of encryption and two stages of
decryption. The first stage of the encryption process maintains the traditional Paillier encryption
process and the second stage process is the encryption using the recommended moduli set by the
RNS Forward converter. At the first stage of the decryption process, our proposed RNS based
reverse converter is adopted and finally, the traditional Paillier decryption process will be used at
the second stage of the decryption process. Figure 1 below demonstrate the proposed
cryptosystem.
Figure 1. Proposed Cryptosystem

5
4.1. The Paillier Cryptosystem
Step 1: Choose two large prime numbers of equal length, p and q
Step 2: Calculate n = p * q
Step 3: Compute
Step 4: Compute
Step 5: Choose . Generator in most general form:
Step 6: Compute
Step 7: Public Key:
Step 8: Private Key:
Encryption:
Step 9: Choose a random integer
Step 10: Plaintext
Step 11: Ciphertext
Decryption:
Step 12: Key:
Step 13: Compute
4.2. The Proposed Algorithm
Step 1: Key: –
– .
Encryption
Step 2: Get the first stage ciphertext (In the usual Paillier
encryption)
Step 3: Input –
Step 4: Compute the second stage ciphertext .
. RNS forward conversion process.
Decryption
Step 4: Key: –
Step 5: Input , the resulting ciphers from Step 4.
Step 6: Compute the first stage decryption process using RNS reverse conversion below:
, the decimal equivalent, now becomes the ciphertext obtained from the Paillier encryption
process.

6
Step 7: Second stage decryption, . In the usual Paillier
decryption
Only the second stage encryption and first stage decryption processes are considered in the
implementation below.
4.3. Stage-2 Encryption Scheme Implementation (Forward Conversion)
At stage-2, the cipher text from stage-1 (Paillier encryption method) has to be encrypted again
using the forward converter with the proposed moduli
set –
– . Each of the ciphertext from stage-1 (Paillier encryption method) would go
through these channels as second layer encryption as follows:
:
Figure 2 below demonstrate the Stage-2 Encryption Scheme Implementation
Figure 2. Stage-2 Encryption Scheme process
4.4. Stage-1 Decryption Process (Reverse Conversion)
The second stage deciphering process of the proposed cryptosystem is accomplished using the
proposed moduli set –
– . The residues (ciphers) in stage-2 encryption stage (forward
conversion) have to be converted to their stage-1 (Paillier encryption method) encryption
cipher(s) (decimal equivalent). This is achieved by using the proposed new reverse converter as
follows:
For two moduli set, we have:

7
is the decimal equivalent.
For three moduli set, we have:
For four moduli set, we have:
For five moduli set, we have:
So the residues (ciphers) in Stage-2 encryption have to be converted back to stage-1 cipher using
the reverse converter for the four moduli set above.
Figure 3 below demonstrate the Stage-1 Decryption implementation.
Figure 3. Stage-1 decryption process

8
Illustration with the Proposed Cryptosystem
Encryption Process (Stage - 2) Using the Moduli Set – .
Let p=13, q=11 then n=143, n2
= 20449, g = 144 and r = 1. The messages 1, 2, 3, 4, 5, 6, 7, 8, 9
and 10 using the stage-one encryption process (Paillier tradition encryption method) have 144,
287, 430, 573, 716, 859, 1002, 1145, 1288, and 1431 as their respective cipher text.
At stage-two, the cipher text from stage-one has to be encrypted again using the forward
converter with the proposed moduli set – . Setting n=4,
our proposed moduli set becomes {24
+ 1, 24
, 24
– 1, 24-1
– 1} = {17, 16, 15, 7}. Each of the cipher
text from stage-one (Paillier traditional encryption method) would go through these channels as
second layer encryption as follows:
:
In summary, we have:
Table 1. Stage – 1 encryption ciphers
Message
m
Stage-1 cipher text
C1
Stage-2 cipher text
C2
1 144 (8,0,9,4)
2 287 (15,15,2,0)
3 430 (5,14,10,3)
4 573 (15,13,3,6)
5 716 (2,12,11,2)
6 859 (9,11,4,5)
7 1002 (16,10,12,1)
8 1145 (6,9,5,4)
9 1288 (13,8,13,0)
10 1431 (3,7,6,3)
Now, C2 are the secured cipher text which will be transmitted.
Decryption Process (Stage - 1) using the Proposed Reverse Converter for Four Moduli set
in section 4.2.
The second stage deciphering process of the proposed cryptosystem is accomplished using the
proposed moduli set – . Setting n=4, our proposed
moduli set becomes {24
+ 1, 24
, 24
– 1, 24-1
– 1} = {17, 16, 15, 7}. The residues (ciphers) in stage-

9
2 have to be converted to their stage-1 encryption cipher(s) (decimal equivalent). So the residues
in table 1, column C2 (Stage-2 cipher text) have to be converted back to stage-1 ciphers as
represented in table 1, column C2 using the proposed new reverse converter with four moduli set
as follows:
For four moduli set, we have:
For , we have :
m1 = 17, m2 = 16, m3 = 15, m4 = 7, r1 = 8, r2 = 0, r3 = 9, r4 = 4.
X = 17.16.15p + v.
Where,
= 144
= 0
X = 17.16.15(0) + 144.
X = 144
:
For , we have:
m1 = 17, m2 = 16, m3 = 15, m4 = 7, r1 = 3, r2 = 7, r3 = 6, r4 = 3.
X = 17.16.15p + v.
Where,
= 1431
= 0
= 0
X = 17.16.15(0) + 1431.

10
X = 1431
5. PERFORMANCE EVALUATION OF THE PROPOSED CRYPTOSYSTEM WITH
THE MODULI SET –
5.1. Key Space Analysis
A key space, also known as a keysapce, is the collection of all valid, feasible, and different keys
in a cryptosystem. The size of the key space is proportional to the security of a cryptosystem.
Because an attacker will try to brute force the message with all conceivable key combinations, an
intercepted communication with a wider keyspace will be more resistant to attackers' decoding
efforts. The more permutations there are, the more secure the encryption scheme becomes. With a
key of length n bits, there are possible keys. As n increases, this number climbs exponentially.
With the propose cryptosystem, the dynamic range, which is dependent on the moduli set, limits
the valid choice of number representation in RNS. When the moduli set is chosen to have a tiny
dynamic range, the algorithm is limited to only a few values that are qualified for stage-2
encryption, making it easy for attackers to crack the system. Hence, the proposed cryptosystem
uses a moduli sets – with dynamic range of 4n-bit, note that, n is the
product of and (Paillier encryption process) which adapt 1024bits key space. Hence our key
space is 4096bits ( which is greater than the Diffie and Hellman "brute force"
attack of 56bits possible keys combination of choices (Diffie & Hellman, n.d.). Because
the moduli set is part of the private component of the classical Paillier cryptosystem, the key
space in the proposed system is increased. Moreover, the proposed moduli set is sufficiently large
enough and best fit design for multiplicative subgroup for the cipher space of the stage-1
encryption ciphers.
Table 2. Key Space Analysis
Key Size
n
Paillier Cryptosystem
Key Space
Proposed Cryptosystem
Key Space
56
128
512
1024
2048
For all of the described techniques, Table 2 displays the probability for brute force attacks, i.e.,
breaking the algorithm through trial and error to obtain the key by utilizing automated software to
make a high number of consecutive guesses. The lower the probability of this attack, the larger
the key space. To obtain a big key space, a good cryptography technique should have a large key
size. The proposed cryptosystem requires exponentially more work ( ) to brute
force attacks which is impractical as compare to the traditional Paillier cryptosystem
( .
5.2. Key Sensitivity Analysis
The goal of key sensitivity analysis is to see how sensitive an encryption method is to changes in
initial conditions. It means that changing the encryption key will result in a completely different
cipher. A good encryption algorithm must be sensitive to the key it uses. To put it another way,

11
changing just one bit of the key must result in an entirely different decoded message than the
original.
RNS with a moduli set has a dynamic range of . For
instance, RNS (3, 5) can represent 15 unique values. Considering the moduli used, the interval [0,
14] is an absolute choice. There exist some ambiguities in any given fixed length number
representation, which is demonstrated in the table 3 below:
Table 3. RNS Representation
From Table 3 above, the interval [0, 5] is an obvious choice for permissible number
representation, which is in row one (1). From column A: 6, 12, 18, 24 . . . as in the order
stated, has equal residue representation as zero (0). From column B: 7, 13, 19, 25 . . . as in the
order stated, has equal residue representation as one (1). From column C: 8, 14, 20, 26 . . . as
in the order stated, has equal residue representation as two (2), likewise column D to F. What
these means is that, without the knowledge of the moduli set (a private key component) in
question or used, one has to guess from infinite numbers which have the same residue
representation as the absolute choice of the required residues, which then becomes a very good
feature for security purposes.
For instance, when (0,0) is transmitted as indicated in table 3, the attacker gets confused as to the
exact number for that residue representation since we have infinite numbers for the same residue
as demonstrated in Column A. This means that, a slight change in the parameters of the moduli
set leads to a dramatic change in the resulting residues. For such a moduli sensitivity, RNS
becomes a strong security parameter because an attacker is left with an unlimited number of
guesses to get the correct number for such residue representation without the knowledge of the
moduli set used which takes infinite years of computations.
Table 4. State of art Results Evaluation
Message
(m)
Stage-one ciphers (
Paillier Encryption
Stage-two ciphers (
Proposed Algorithm
1 144 (8,0,9,4)
2 287 (15,15,2,0)
3 430 (5,14,10,3)
4 573 (12,13,3,6)
5 716 (2,12,11,2)
6 859 (9,11,4,5)
7 1002 (16,10,12,1)
8 1145 (6,9,5,4)
9 1288 (13,8,13,0)
10 1431 (3,7,6,3)

12
In the table 4 above, transmitting all the plaintexts (messages) are secured because the second
level of encryption (proposed algorithm) will transform into . This resolves the security
vulnerabilities of the Paillier cryptosystem asserted in (Asiedu & Salifu, 2020), making the
scheme fully functional in real world applications such as e-voting systems, e-cash systems and
its related applications.
6. CONCLUSION
An improved Paillier cryptosystem have been implemented based on the Residue Number
System. The improved cryptosystem of the tradition Paillier cryptosystem have two stages of
encryption and decryption. The first stage is the traditional Paillier encryption process and the
second stage is to pass the cypher text obtained from Paillier encryption process into moduli
(forward conversion) to prevent the computation of the plaintext from the cipher text without
solving its security assumption of Decisional Composite Residuosity or the knowledge of its
private keys under mathematical attacks. The key length is also enhanced with the key space of
4096-bits ( as the moduli are part of the private key component of the proposed
cryptosystem. The security of the cryptosystem is proportional to the length of the private key.
This will help reduce the vulnerability to attacks like brute force. The key sensitivity analysis
offers strong resistance to Brute-force and key sensitivity attacks.
7. FUTURE DIRECTIONS
 Further research is of interest in computational time and cost since the proposed
cryptosystem focused on improving the security robustness of the Paillier Cryptosystem.
 Since RNS is demonstrating the very promising security robustness of Paillier
Homomorphic Encryption Scheme, similar applications are recommended for other
public-key cryptosystem exhibiting security threats.
 How homomorphic computation (additive or multiplicative) could be carried out after the
stage-two encryption phase is recommended for future work.
REFERENCES
[1] Albugmi, A., Alassafi, M. O., Walters, R., & Wills, G. (2016). Data security in cloud computing. 5th
International Conference on Future Generation Communication Technologies, FGCT 2016, August,
55–59. https://doi.org/10.1109/FGCT.2016.7605062
[2] Asiedu, D., & Salifu, A.-M. (2020). Security Evaluation of Pailiar Homormophic Encryption
Scheme. Asian Journal of Research in Computer Science, 6(3), 12–17.
https://doi.org/10.9734/ajrcos/2020/v6i330159
[3] Catalano, D., Gennaro, R., & Howgrave-Graham, N. (2001). The bit security of paillier’s encryption
scheme and its applications. Lecture Notes in Computer Science (Including Subseries Lecture Notes
in Artificial Intelligence and Lecture Notes in Bioinformatics), 2045, 229–243.
https://doi.org/10.1007/3-540-44987-6_15
[4] Damgård, I. B., Jurik, M. J., Brics, M. J. J., Damgård, I., & Jurik, M. (2003). A length-flexible
threshold cryptosystem with applications. Springer. https://link.springer.com/chapter/10.1007/3-540-
45067-X_30
[5] Damgård, I., Jurik, M., & Nielsen, J. B. (2010). A generalization of Paillier’s public-key system with
applications to electronic voting. International Journal of Information Security, 9(6), 371–385.
https://doi.org/10.1007/S10207-010-0119-9
[6] Diffie, W., & Hellman, M. E. (n.d.). Rivest 2014 L14.1 Paper New Directions in Cryptography
Invited PapDiffie, W., & Hellman, M. E. (n.d.). Rivest 2014 L14.1 Paper New Directions in
Cryptography Invited Paper. 29–40.er. 29–40.
[7] El Makkaoui, K., Ezzati, A., Beni-Hssane, A., & Ouhmad, S. (2020). Fast Cloud–Paillier
homomorphic schemes for protecting confidentiality of sensitive data in cloud computing. Journal of

13
Ambient Intelligence and Humanized Computing, 11(6), 2205–2214. https://doi.org/10.1007/S12652-
019-01366-3
[8] Flores, I. (1969). Residue Arithmetic and Its Application to Computer Technology (Nicholas S.
Szabo and Richard I. Tanaka). SIAM Review, 11(1). https://doi.org/10.1137/1011027
[9] Fontaine, C., & Galand, F. (2007). A survey of homomorphic encryption for nonspecialists. Eurasip
Journal on Information Security, 2007. https://doi.org/10.1155/2007/13801
[10] Gupta, C. P., & Sharma, I. (2013). A fully homomorphic encryption scheme with symmetric keys
with application to private data processing in clouds. 2013 4th International Conference on the
Network of the Future, NoF 2013, August 2018. https://doi.org/10.1109/NOF.2013.6724526
[11] Gupta Iti Sharma Asso, C. P. (2013). Fully Homomorphic Encryption Scheme with Symmetric Keys.
http://arxiv.org/abs/1310.2452
[12] Jiang, C., & Pang, Y. (2020). Encrypted images-based reversible data hiding in Paillier cryptosystem.
Multimedia Tools and Applications, 79(1–2), 693–711. https://doi.org/10.1007/S11042-019-07874-W
[13] Jurik, M. J. (2003). Extensions to the paillier cryptosystem with applications to cryptological
protocols. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.67.9647&rep=rep1&type=pdf
[14] Moore, C., O’Neill, M., … E. O.-… I. I., & 2014, undefined. (2014). Practical homomorphic
encryption: A survey. Ieeexplore.Ieee.Org, 2792–2795. https://doi.org/10.1109/ISCAS.2014.6865753
[15] Moulay, D., Ouadghiri, E., Hassan, N., Ibtihal, M., & Driss, E. O. (2017). Homomorphic encryption
as a service for outsourced images in mobile cloud computing environment. Igi-Global.Com.
https://doi.org/10.4018/IJCAC.2017040103
[16] Paillier, P. (1999). Public-Key Cryptosystems Based on Composite Degree Residuosity Classes. BT -
Advances in Cryptology - EUROCRYPT ’99, International Conference on the Theory and
Application of Cryptographic Techniques, Prague, Czech Republic, May 2-6, 1999, Proceeding. 223–
238. https://doi.org/10.1007/3-540-48910-X_16%0Ahttps://www.wikidata.org/entity/Q56287504
[17] Papisetty, S. (2017). Homomorphic Encryption: Working and Analytical Assessment: DGHV, HElib,
Paillier, FHEW and HE in cloud security. https://www.diva-
portal.org/smash/record.jsf?pid=diva2:1082551
[18] Pettersen, N., & Gjøsteen, K. (2016). Applications of Paillier s Cryptosystem.
https://ntnuopen.ntnu.no/ntnu-xmlui/bitstream/handle/11250/2410268/15986_FULLTEXT.pdf
[19] Schoinianakis, D. (2020). Residue arithmetic systems in cryptography: a survey on modern security
applications. Journal of Cryptographic Engineering, 10(3). https://doi.org/10.1007/s13389-020-
00231-w
[20] Shihab Ahmed, H. A., & Zolkipli, M. F. (2016). Data Security Issues in Cloud Computing: Review.
International Journal of Software Engineering and Computer Systems, 2(February), 58–65.
https://doi.org/10.15282/ijsecs.2.2016.5.0016

Secured Paillier Homomorphic Encryption Scheme Based on the Residue Number System

More Related Content

Similar to Secured Paillier Homomorphic Encryption Scheme Based on the Residue Number System (20)

More from ijcisjournal2 (11)

Recently uploaded (20)

Secured Paillier Homomorphic Encryption Scheme Based on the Residue Number System