Content uploaded by Maiwan B. Abdulrazzaq
Author content
All content in this area was uploaded by Maiwan B. Abdulrazzaq on Mar 15, 2020
Content may be subject to copyright.
JZS (2019) 21–1 (Part-A)
99
Selective Multi Keys to Modify RSA Algorithm
Maiwan Bahjat Abdulrazzaq
Dept. of Computer Science, Faculty of Science, University of Zakho, Kurdistan Region – Iraq.
E-mail:maiwan.abdulrazzaq@uoz.edu.krd
Article info
Abstract
Original: 3 November
2018
Revised: 10 January 2019
Accepted: 17 February
2019
Published online: 20 June
2019
The use of the internet and communication technology noticeably contributes to
development in all branches of science. The data that is transferred to communication
channel are unsecured. Cryptography including security and integrity is a domain that
provides policies for having privacy of the data. Protection of the data transferred through
the internet is very important since such data when transferred through an unprotected
channel may be attacked by a third party. RSA represents one of the public key
cryptography methods that generate the modulus using two primes. These two primes
represent the weakness of RSA benefited from and used by attackers. This paper proposes a
new modified RSA method using selective multi keys encryption and decryption for the
same modulus instead of using two keys, one for encryption and the other for decryption.
This new method guarantees and increases the security of RSA.
Key Words:
Cryptography, public key
cryptography, RSA
algorithm, asymmetric
key, symmetric key.
I. Introduction
The development of network and communication technology stands behind the revolution of information
technology. Security is one of the problems that permanently exist in network and communication
technology. This problem calls for and simultaneously demands the development of a new strong
cryptography method. Cryptography falls into two main types, namely symmetric key and asymmetric key.
In symmetric cryptography, the same key is used for encryption and decryption, while in asymmetric
cryptography the key used for encryption differs from that used for decryption. The concept of asymmetric
key encryption cryptosystem is also called public key cryptography was introduced by Whitfield Diffe and
Martin Hellman in 1976[1]. Many methods of public key cryptography have been developed, viz. Rabin
cryptosystem[2], Elagmal cryptosystem [3], ECC [4]. The most common one in use is RSA which was
introduced in 1978 by (Rivest Shamir Adleman) [5]. The encryption key e and the modulus n are quite
public in RSA. The Privacy in RSA is represented by the decryption key d with the Euler function. Currently
RSA is the safest technique and the most effective method, yet has certain drawbacks represented mainly by
the possibility of coming under Discrete Logarithm, Cycle, Brute Force, Mathematical, and Timing attack
[6, 7]. The RSA cryptosystem protocol involves two prime numbers to produce the modulus n. These two
prime numbers weakness form the source of the weakness of RSA. The larger prime number gives higher
security of the algorithm. The attacker’s main goal is to break the security of this algorithm. Such a break can
be done by factorizing the modulus which takes place exponential time. This operation can be adopted in
order to discover the value of the modulus [8]. There are many researchers work in deferent ways to
modified RSA the following papers some of them:
R. S. Dhakar et al. (2012) presented a new cryptography algorithm depending on the properties of the
additive homomorphic. The new method is called Modified RSA Encryption Algorithm (MREA). MREA is
secure for both the decisional composite residuosity, i.e. the intractability hypothesis assumptions as well as
Journal homepage www.jzs.univsul.edu.iq
Journal of Zankoy Sulaimani
Part-A- (Pure and Applied Sciences)
JZS (2019) 21–1 (Part-A)
100
factoring problem compared to RSA. The additive homomorphism cryptosystem scheme public-key of m1
and m2, can compute the encryption of m1 + m2 to improves the security [9].
K. Somsuk (2016) proposed a new method, namely d-RSA to reduce the computation of the decryption
process. The method uses a new private key not the traditional inverse. The new private key must have low
Hamming weight. Also, the result of the multiplication between public key and private key modulo Euler
function must be small. It is worthy to note that the decryption process of d-RSA is more efficient than RSA
when the size of n is large [10].
P. Chaudhury, et al. (2017) proposed a modified RSA cryptosystem algorithm called “Asymmetric key
based Cryptographic Algorithm using Four Prime numbers to secure message communication (ACAFP)”
to handle four prime numbers used to generate the modulus and the Euler function. ACAFP can solve the
factorization problem of RSA by including smallest prime numbers and hence the scheme becomes more
advanced as far as memory consumption and computational speed are concerned [11].
I. G. Amalarethinam and H. Leena (2017) proposed the Enhanced RSA (ERSA) algorithm that uses two
additional prime numbers in the Standard RSA algorithm. The four prime numbers are used to generate (N1,
N2). N1 is used with encryption key e, while N2 is used with decryption key d. This idea was raised the high
Speed and security RSA algorithm which use two random prime numbers for the key generation
process[12].
A. Goel (2017) proposed algorithm use double encryption and decryption that’s mean use double private
and double public keys for dual modulus to provide security against Brute-force attacks. The basic
motivation behind the carrying out of this research is that can factor modulus (n) into its prime numbers in
conventional RSA algorithm, to generate a private key. As such, to remove this weakness, the dual modulus
and substantially improve the security of the system [13].
I. Yakymenko et al. (2018) developed the scheme of modular multiplication and modular exponentiation
algorithms. This was done by replacing the multiplication operation with the addition operation.
Consequently, the process of encryption / decryption of information were accelerated. The developed method
also reduced the temporal complexity of modular exponentiation in comparison with the classical one and
increased the speed of the RSA algorithm[14].
As shown the goal of the researchers is to increase the security and the complexity of the RSA algorithm.
The researchers aim is to complicate the mission of decipher the message from the hackers. Some of them
increased the number of primes that generate the modulus. Other goes though the multiplication or raising
the power of the keys for encryption and decryption. The methodology of this paper tries another way
through using multi key encryption and decryption as much as the numbers of message blocks that can be
selected from the candidate keys.
The remaining of this paper is organized as follows. In Section II, RSA algorithm is presented. In Section
III, light is shed on the modified RSA key selective. Section IV puts forward the results of the key
generations. Finally, Section V presents the conclusions.
II. RSA ALGORTHM
The RSA algorithm can be comprises three main steps, namely key generation, encryption and
decryption[15]:
1. Key generation
i. Select two large prime and
ii. Compute the modulus
iii. Calculate the Euler function as:
φ
iv. Choose the public key ‘e’ according to
a. φ
b. φ
v. Find private key d such that
JZS (2019) 21–1 (Part-A)
101
vi. The public and the privet
2. Encryption
i. The message , coded to integer such that:
ii. Cipher the message by
3. Decryption
i. Recover the message by
III. PROPOSED SELECTIVE MULTI KEYS FOR MODIFYING RSA ALGORTHEM
Selective Multi Keys for modifying RSA Algorithm consists of three steps, namely key generation,
encryption and decryption:
1. Key generation
i. Select two large prime and
ii. Compute the modulus
iii. Calculate the Euler function as:
iv. Put all primes less than in candidate vector (CK) :
!" #$"%&' {
" &
(
&
(
&
(
'
&
(
&
(
'
)
)
v. Finding the encryptions and decryptions keys:
* '
!" #$"%&
(
!+ #$"%&
,
" &
(
&
(
-
.
&
(
;
/
.
&
,
'
* * 0 '
)
)
)
vi. The Public keys are-
.
.
vii. The Private keys are /
.
2. Encryption:
i. Split the messages to blocks, which each block
1
is coded to integer less than n
ii. Select randomly number of keys
1
from-
.
and
1
from/
.
equal to the number of
1
.
iii. !" #$"%
1
2
3
34
5
67) ;
1
3. Decryption:
!" #$"%
1
3
2
38
5
67)
Example1 of Selective Multi Keys RSA Algorithm
1. Key generation
i. Let p = 7 and q = 13
ii. Compute the modulus 9 : ;
iii. Calculate the Euler function φ9 : 9<
JZS (2019) 21–1 (Part-A)
102
iv. CK= (2 ,3 ,5 ,7 ,11 ,13 ,17 ,19 ,23 ,29 ,31 ,37 ,41 ,47 ,53 ,59 ,61 ,67 ,71 ).
v. The keys (=
>
?
>
) that satisfy the condition from the candidate keys (CK) are :
1) =
@
A'?
@
<;
2) =
B
9'?
B
:
3) =
C
'?
C
A;
4) =
D
:'?
D
E
5) =
F
<:'?
F
G9
vi. Public keys are (5, 7, 11, 13, 23), 91) and Private keys are (29, 31, 59, 61, 47), 72)
2. Encryption
i. Divide the messages(GO) to blocks; each block code less than (91), then G =07 and O =15
ii. Select any two keys (
C
F
) which they are (11, 47) from the public key sequence, and the inverse of
them are ()
C
)
F
) which they are (23, 47) .
1) 2
@
9
@@
67); <H
2) 2
B
A
BC
67); HA
3. Decryption
1)
@
<H
FI
67); 9
2)
B
HA
DJ
67); A
The number of primes that can be consider as candidate keys (CK) are 19 which there values less than φ (n)=
72. From these 19 primes, the numbers that satisfy the conditions in this example are 10. They are split into
two groups. Group1: in (=
>
) consists of five prime’s (5, 7, 11, 13, and 23) they will be use for encryption
selecting. Group 2: in (?
>
) which also consists of five primes (29, 31, 59, 61, and 47) they will be use for
decryption. The decryption keys selection depends on the encryption key selection. For encrypting the
messages, (GO) should be divided into two blocks. Then, each block is coded to integer. The message block
length has to be less than (91). Accordingly, G will be (7) and O will be (15). Because they are two blocks,
two keys need to be chosen for encryptions. Let be (
C
F
) and in the same sequence for decryption ()
C
)
F
).
These keys are then used to apply the encryption and decryption processes.
Example 2 of Selective Multi Keys for modifying RSA Algorithm
1. Key generation
i. Let p = 1009 and q = 1901
ii. Compute the modulus ; ; ;H;
iii. Calculate the Euler function φ ; ; ;A<
iv. CK= ( 2 ,3 ,5 ,7 ,11 ,13 ,17 ,19 ,23 ,29 ,31 ,…, 1915163 ,1915183 ).
v. The encryption and decryption keys are:
1)
@
:')
@
9E9H99
2)
B
<:')
B
AH<HH9
3)
C
G:')
C
<G99
48819)
DKK@I
;A;')
DKK@I
EH999;
48820)
DKKBL
;AH:')
DKKBL
<:;<G9
vi. Public keys are ((13, 23, 43… 1915019, 1915183), 1918109).
vii. Private keys are ((1767877, 582887, 1247107,…, 687779, 1239247), 1009 , 1901, 1915200)
2. Encryption
i. Divide the messages (SEND HELP SOON) to blocks; each block code length is less than (n=
1918109). The message becomes four blocks
JZS (2019) 21–1 (Part-A)
103
SEN=180513, DHE=30704, LPS=111518, OON=141423.
ii. Select any four keys
BBJL
BMLB
BMIBB
CBMKM
from the public key sequence with values (66643,
77773, 1000003, 1915183) used for cipher the message and the inverse of selected keys are
)
BBJL
)
BMLB
)
BMIBB
)
CBMKM
with values (66643, 77773, 1000003, 1915183) used for decipher
the message:
1) 2
@
HA:
MMMDC
67);H; :<HA
2) 2
B
:9G
JJJJC
67);H; ;HE9
3) 2
C
AH
@LLLLLC
67);H; 9GHGGGH
4) 2
D
GG<:
@I@F@KC
67);H; A:G<
3. Decryption
1)
@
:<HA
@FMICLJ
67);H; HA:
2)
B
;HE9
MLLCJ
67);H; :9G
3)
@
9GHGGGH
@BFBBMJ
67);H; AH
4)
@
A:G<
@BCIBDJ
67);H; GG<:
The number of primes that can be consider as candidate keys (CK) are 143082 which there values are
less than φ (n)= 1915200. Prime numbers that satisfy the conditions in the example are 48820. They are split
into two groups. Group 1 consists of (71541) primes (13, 23, 43… 1915019, 1915083) that are used for
encryption. Group 2 also comprises (71541) primes (1767877, 582887, 1247107, 687779, and 1239247)
which used for decryption. The decryption keys selection depends on the encryption keys selection. For
encrypting the messages, (SEND HELP SOON) is divided into blocks; each block code length is less than
(n) which is value equal to (1918109), then SEN=180513, DHE=30704, LPS=111518, ON=141423. The
numbers of blocks are four. This means that for encryptions, four keys need to be selected (66643, 77773,
1000003, and 1915183). This enforces the choice of the decryption keys (1569307, 60037, 1252267, and
1239247). These keys are then used for the application of the encryption and decryption processes.
V. RESULTS OF KEY GENERATION
Selecting the two prime numbers p and q is very important for RSA. The two primes will determine the
size of the modulus (n) and Euler function φ (n).The modulus will determine the size of the message blocks,
while the Euler function will determine the number and size of the keys. The last raw in table 1 shows that
the values of selecting p equal to 1129, q equal to 8887 determine the value of the Euler function to be equal
10023408.The number of primes they can be consider as candidate keys are 666045.The keys that satisfy the
condition from candidate keys are 135454 as shown in Table1.can select keys from them for encryption and
decryption as much as the block messages have.
Table-1: The number of keys produced from selecting (p and q).
p q φ (n) n-primes n-keys
7 13 72 20 10
17 43 672 121 64
79 97 7488 948 388
223 433 95904 9242 2694
353 929 326656 28137 25392
1901 1009 1915200 143082 48800
1129 8887 10023408 666045 135454
The relationship between the digit number of Euler function and the number of keys is increasing as
shown in Figure 1.
JZS (2019) 21–1 (Part-A)
104
Figure-1: The number of Euler function digit and number of keys.
VI. CONCLUSIONS
The original RSA algorithm process depends of using two keys. The first key called public key which is
use for ciphering the blocks of the messages. The second key which is the privet key is use for recovering the
ciphered blocks of the messages. That is mean this algorithm depend on only one key for deciphering the
message. As known there are many methods used for attacking RSA algorithm. Therefore the solution that
RSA is used to solve this problem is by increasing the number of the digits of the keys and the modulus. This
solution raises the difficulty to be attacked easily. The current paper presents proposes method for modifying
RSA cryptography algorithm that depends on using multi keys, instead of using only one key for decryption.
Each block of the message uses its own selected key. The key will use exactly one time. Even finding one
privet key from the attacker will not recover all messages but only recover that block, that is mean
increasing the security. The chosen of the two large prime’s p and q gives a larger Euler function that leads
to a larger numbers of the keys that used for encryption and decryption; as a result led to increase the
complexity. It is very difficult for any hackers to guess the numbers of the keys have been used for
encryption with the size of the block also which keys have been selected and in any sequence they used. The
limitation of the proposed method is the calculation time. Which will increase depends on (p, q) and the
numbers of selected keys (e, d).
References:
[1] A. K. Mishra and S. G. Samaddar, "Generation of symmetric sharing key using ABC conjecture",
International Conference on Energy, Communication, Data Analytics and Soft Computing (ICECDS),
2017, pp. 144-150. (2017).
[2] M. Elia, M. Piva, and D. Schipani, "The Rabin cryptosystem revisited", Applicable Algebra in
Engineering, Communication and Computing. Vol. 26, pp. 251-275. (2015).
[3] M.-S. Hwang, C.-C. Chang, and K.-F. Hwang, "An ElGamal-like cryptosystem for enciphering large
messages", IEEE Transactions on Knowledge and Data Engineering. Vol. 14, pp. 445-446. (2002).
[4] M. Rosing, "Implementing elliptic curve cryptography", Manning Greenwich, (1999).
[5] S. J. Aboud, M. A. AL-Fayoumi, M. Al-Fayoumi, and H. S. Jabbar, "An efficient RSA public key
encryption scheme", in Information Technology: New Generations 2008. Fifth International Conference
on, 2008, pp. 127-130. (2008).
[6] F. F. Moghaddam, S. D. Varnosfaderani, I. Ghavam, and S. Mobedi, "A client-based user authentication
and encryption algorithm for secure accessing to cloud servers based on modified Diffie-Hellman and
RSA small-e", in Research and Development (SCOReD), IEEE Student Conference on, 2013, pp. 175-
180. (2013).
[7] V. Shende and M. Kulkarni, "FPGA based hardware implementation of hybrid cryptographic algorithm
for encryption and decryption", in Electrical, Electronics, Communication, Computer, and Optimization
Techniques (ICEECCOT), International Conference on, 2017, pp. 416-419. (2017).
[8] M. Lakkadwala and S. Valiveti, "Parallel generation of RSA keys—A review", in Cloud Computing,
Data Science & Engineering-Confluence, 7th International Conference on, 2017, pp. 350-355. (2017).
0
50000
100000
150000
1234567
n-Digit n-keys
JZS (2019) 21–1 (Part-A)
105
[9] R. S. Dhakar, A. K. Gupta, and P. Sharma, "Modified RSA encryption algorithm (MREA)", Second
International Conference on Advanced Computing & Communication Technologies, 2012, pp. 426-429.
(2012).
[10]K. Somsuk, "The improving decryption process of RSA by choosing new private key", in Information
Technology and Electrical Engineering (ICITEE), 8th International Conference on, 2016, pp. 1-4.
(2016).
[11]P. Chaudhury, S. Dhang, M. Roy, S. Deb, J. Saha, A. Mallik, et al., "ACAFP: Asymmetric key based
cryptographic algorithm using four prime numbers to secure message communication. A review on RSA
algorithm", in Industrial Automation and Electromechanical Engineering Conference (IEMECON), 8th
Annual, 2017, pp. 332-337. (2017).
[12]I. G. Amalarethinam and H. Leena, "Enhanced RSA Algorithm with Varying Key Sizes for Data Security
in Cloud", in Computing and Communication Technologies (WCCCT), World Congress on, 2017, pp.
172-175. (2017).
[13]A. Goel, "Encryption algorithm using dual modulus", in Computational Intelligence & Communication
Technology (CICT), 3rd International Conference on, 2017, pp. 1-4. (2017).
[14]I. Yakymenko, M. Kasianchuk, S. Ivasiev, A. Melnyk, and Y. M. Nykolaichuk, "Realization of Rsa
cryptographic algorithm based on vector-module method of modular exponention", in Advanced Trends
in Radioelecrtronics, Telecommunications and Computer Engineering (TCSET), 14th International
Conference on, 2018, pp. 550-554. (2018).
[15]T. C. Segar and R. Vijayaragavan, "Pell's RSA key generation and its security analysis", in Computing,
Communications and Networking Technologies (ICCCNT), Fourth International Conference on, 2013,
pp. 1-5. (2013).
JZS (2019) 21–1 (Part-A)
106
