INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue X October 2025
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An Improved Hybrid Encryption Scheme Based on the Sequence of
Reduced Residue Systems
Muhammad A. H.
1
, Ibrahim A. A.
2
, Garba A. I.
2
, Sarki M. N.
3
, Mua’azu S. B.
3
, Abubakar S. F.
3
, Shehu
S.
4
, James T. O.
3
and Abubakar T. U.
5
1
Department of Science, Mathematics Unit, State Collage of Basic & Remedial Studies, Sokoto.
2
Department of Mathematics, UsmanuDanfodio University Sokoto.
3
Department of Mathematics, Abdullahi Fodio University of Science and Technology, Aliero.
4
Department of Mathematics, Sokoto State University.
5
Department of Mathematics, ShehuShagari College of Education, Sokoto.
DOI: https://dx.doi.org/10.51244/IJRSI.2025.1210000348
Received: 10 November 2025; Accepted: 16 November 2025; Published: 22 November 2025
ABSTRACT
The rapid growth of digital communication has created an urgent demand for advanced cryptographic
techniques that ensure both security and efficiency. As reliance on digital platforms increases, so does the risk
of cyber threats and data breaches. Robust cryptography is, therefore, essential to protect sensitive information
and maintain trust in digital transactions. This research proposes an improved composed hybrid cryptosystem
that integrates Transposition, Caesar, and Hill ciphers, followed sequentially by RSA encryption. The study
examines how a hybrid of four ciphers can be attacked when treated as a composite function. To further
enhance security, a sequence of Reduced Residue System (RRS) values was introduced to replace ASCII
characters after the third cipher (Hill cipher), adding an additional layer of residue-based encryption before the
final RSA stage. The findings demonstrate that the improved hybrid cryptosystem significantly enhances data
security and key generation efficiency, adding a new level of complexity that makes it more challenging for
attackers to guess or compute decryption keys.
Keywords: Hybrid, Encryption, Decryption, Attacks and RSA.
INTRODUCTION
Cryptography is the study of secure communication techniques that allow the sender and intended recipient of
a message to view and understand its contents, (Ibrahim, et al., 2021). In today's digital world, the need for
robust data security has never been more critical as now. Information exchange proliferates across various
platforms and networks, safeguarding sensitive data from unauthorized access, modification, or disclosure
becomes paramount, (Shehu et al., 2023). Individual encryption methods, while effective, often face challenges
related to computational efficiency, key management, and vulnerable to attacks, (Yao & Su, 2021). Hybrid
encryption systems have emerged as a widely adopted solution to these challenges, (Manna, et al., 2017).
Recently, numerous research endeavours have focused on enhancing security using the hybrid cryptosystem.
For instance, Rufa’i et al., (2020) integrated RSA, Shifting, and Hill ciphers to enhance security and
robustness, while Hassan, Garko, et al., (2023) combined Hill and Transposition ciphers to improve data
protection. Despite these advancements, vulnerabilities remain in existing hybrid cryptosystems. This paper
aims to enhance data security by improving a hybrid encryption system that sequentially integrates
Transposition, Caesar, Hill, and RSA ciphers, using a sequence of Reduced Residue Systems (RRS) to replace
ASCII characters.
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Integrating an RRS sequence into a hybrid cryptosystem enhances both data security and efficiency by
combining the strengths of hybrid encryption and residue-based techniques. This modification addresses
vulnerabilities found in the existing system and strengthens its resistance to potential attacks. The additional
layer of complexity introduced by RRS makes it considerably more difficult for attackers to decipher
ciphertexts.
LITERATURE REVIEW
The concept of hybrid cryptography has attracted considerable attention in recent years due to its ability to
provide enhanced security features. Some of the most relevant works are:
Rufa’i et al. (2020) combined three ciphers—RSA, Shifting, and Hill—and represented them as bijective
functions. The composition of these functions resulted in an improved hybrid cipher system with enhanced
encryption performance.
Khan, Pradhan, & Chandavarkar (2021) proposed a hybrid algorithm based on the Guillou–Quisquater scheme
and RSA. Their model aims to ensure data integrity through RSA-based key generation, while the Guillou–
Quisquater component handles integrity and confidentiality.
Suhasini and Bushra (2021) introduced a three-level encryption technique designed to overcome the limitations
of single-key encryption by combining the Advanced Encryption Standard (AES), Data Encryption Standard
(DES), and RSA. This multi-tier approach provides greater resistance against brute-force and key-compromise
attacks.
Saja, Zaynab, and Jaafer (2023) proposed a two-layer hybrid cryptosystem. In the first layer, plaintext is
encrypted using a modified Playfair cipher to produce ciphertext, which is subsequently re-encrypted using
RSA to generate the final ciphertext. This layered design enhances overall data confidentiality.
Prakash, Saeed, Rajan, Mohammad, and Ahmed (2023), in their paper presented a scheme combining RSA
with a Simple Symmetric Key (SSK) algorithm.
Susmitha, Kumar, and Bulla (2023) demonstrated that a hybrid cryptographic approach combining AES and
ElGamal enhances file security by integrating symmetric and asymmetric techniques. Their method ensures
confidentiality and integrity for stored data and provides valuable insights into key management strategies.
Hassan et al. (2023) enhanced data security using a combination of Hill and Transposition ciphers. In their
method, plaintext is first encrypted with the Hill cipher and then re-encrypted with the Transposition cipher,
resulting in a more complex ciphertexts.
Ariffin, Wijonarko, Suwarno, and Kristianto (2024) combined the Unimodular Hill Cipher with RSA to
produce a more secure text encryption system.
Despite these significant contributions, hybrid cryptosystems remain susceptible to certain forms of attack.
Consequently, there is a need for further improvement. This study aims to advance the composed hybrid
cryptosystem by incorporating a sequence of Reduced Residue Systems (RRS) to replace ASCII characters,
thereby increasing security complexity.
METHODOLOGY
The methodology employed in this study incorporates several fundamental concepts of number theory,
including the Greatest Common Divisor (GCD), Euler’s Totient Function , Congruence, and the concept
of the Reduced Residue System (RRS), as well as the American Standard Code for Information Interchange
(ASCII) table. Furthermore, it integrates four cryptographic techniques: Transposition cipher, Caesar cipher,
Hill cipher, and RSA algorithm, alongside the method of composing multiple ciphers.
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ASCII Table
The complete 7-bit ASCII table and corresponding decimal equivalents are presented in table 1.
Table 1: ASCII Table (Google Search, 2024)
Dec
Chr
Dec
Chr
Dec
Chr
Dec
Chr
Dec
Chr
0
NUL
26
SUB
52
4
78
N
104
h
1
SOH
27
ESC
53
5
79
O
105
i
2
STX
28
FS
54
6
80
P
106
j
3
ETX
29
GS
55
7
81
Q
107
k
4
EOT
30
RS
56
8
82
R
108
l
5
ENQ
31
US
57
9
83
S
109
m
6
ACK
32
58
:
84
T
110
n
7
BEL
33
!
59
;
85
U
111
o
8
BS
34
"
60
<
86
V
112
p
9
HT
35
#
61
=
87
W
113
q
10
LF
36
$
62
>
88
X
114
r
11
VT
37
%
63
?
89
Y
115
s
12
FF
38
&
64
@
90
Z
116
t
13
CR
39
'
65
A
91
[
117
u
14
SO
40
(
66
B
92
\
118
v
15
SI
41
)
67
C
93
]
119
w
16
DLE
42
*
68
D
94
^
120
x
17
DC1
43
+
69
E
95
_
121
y
18
DC2
44
,
70
F
96
`
122
z
19
DC3
45
-
71
G
97
a
123
{
20
DC4
46
.
72
H
98
b
124
|
21
NAK
47
/
73
I
99
c
125
}
22
SYN
48
0
74
J
100
d
126
~
23
ETB
49
1
75
K
101
e
127
DEL
24
CAN
50
2
76
L
102
f
25
EM
51
3
77
M
103
g
Fundamental Concepts of Number Theory
This section presents key number theory concepts utilized in the study.
Greatest Common Divisor (GCD)
If m and n are integers, a positive integer d is the GCD of m and n if d divides both m and n, and d is the
greatest among all common divisors of m and n. For instance, the GCD of 24 and 60 is 12, denoted as (24, 60)
= 12.
Relatively Prime Integers
Two or more integers are said to be relatively prime if their GCD is 1, (Hardy & Wright, 2008). For instance,
the factors of 8 are {1, 2, 4, 8}, and the factors of 15 are {1, 3, 5, 15}. Thus, GCD (8, 15) = 1, meaning 8 and
15 are relatively prime.
Euler's Totient Function
Euler's Totient Function, denoted by , counts the number of positive integers less than or equal to N that
are relatively prime to N, (Rosen, 2011). Mathematically,
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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, such that GCD, .
For instance: , as numbers less than or equal to are: , for which integers
≤ 10 that are coprime with 10 are: 1, 3, 7, 9.
Similarly, , because the integers ≤ 12 that are coprime with 12 are {1, 5, 7, 11}.
It follows that
thus, in general
, where is prime.
Properties of
The Euler Totient function has many useful properties:
(i) is multiplicative when
thus,
For instance,
(ii) For any prime p,
, hence,
if
then,
, (Rosen, 2011)
Congruence
If and are integers and is a positive integer, then is said to be congruent to modulo if divides
. This is written as , (Rosen, 2011).
For instance, , since divides .
Least Residue
If and is the remainder when dividing n by m then is called the least residue of modulo , (Rosen,
2011).
Residue Classes
A residue class modulo is a set of integers that are congruent to each other modulo . Each residue class
contains exactly one integer from . For modulo 4, there are four residue classes
where
Complete Residue System
A complete residue system modulo is any set of integers
such that
, (Rosen, 2011).
Reduced Residue System (RRS)
A Reduced Residue System (RRS) modulo is a set of integers that represent all possible residue classes
modulo that are relatively prime to (Rosen, 2011).
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A set S is an RRS modulo m if:
i)
(ii)
(iii) GCD
The set forms an RRS modulo 9; since
(i)
and
(ii)
(iii)
,
3.3 Composition of Four Ciphers
The four ciphers: Transposition, Caesar, Hill, and RSA were modelled as mathematical functions and
composed sequentially using the concept of function composition.
Encryption Process
Step I: Transposition Cipher () - Rearrange the plaintext according to a specific key to obtain ciphertexts
as:
that is
Step II: Caesar Cipher () - Shift each letter in
by a fixed number
of positions down the ASCII table
to obtain
as:
that is
.
Step III: Convert
into its ASCII numerical equivalents, denoted
.
Step IV: Hill Cipher (): Encrypt
using an invertible key matrix
to obtain
as:
that is
,
Step V: RSA Encryption () - Encrypt
using modular exponentiation with public key to obtain
as:
that is
Thus, the composed cipher is:
Where:
ciphertexts from Caesar encryption
ASCII numerical equivalent of
ciphertexts from Hill encryption
ciphertexts from RSA encryption
the RSA modulus
Decryption Process
Decryption reverses the encryption steps as follows:
Step I: RSA Decryption - decrypt
using private key to obtain
as:
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Step II: Hill Decryption - Multiply
inverse key matrix
to recover
as:
Step III: Convert
to ASCII characters to obtain
Step IV: Caesar Decryption - shift each character of
upward by
positions:
Step V: Transposition Decryption - rearrange
back to its original order using the transposition key
:
Hence, the complete decryption process is represented as:
Where:
RSA decryption function
Hill decryption function
Caesar decryption function
Transposition decryption function
inverse of Hill key matrix
ciphertexts of the composed system
RESULT AND ANALYSIS
This section presents a comprehensive analysis of the findings, highlighting the performance and security
enhancements achieved through the proposed approach.
Cryptanalysis of the Combined Four Ciphers
If an attacker knows the cipher order (Transposition → Caesar → Hill → RSA), the hybrid can possibly be
attacked by peeling the layers in reverse: first break RSA, then the Hill cipher, then Caesar, and finally the
transposition, as illustrated below:
Step I: Apply RSA cryptanalytic methods to recover the decrypted block stream
Step II: Use Hill-specific attacks (known-plaintext, linear algebraic key recovery, or brute force) on the result
to recover the message before Hill.
Step III: Break the Caesar shift via frequency analysis, known plaintext, or simple brute force to get the pre-
Caesar text.
Step IV: Determine and reverse the transposition pattern to reconstruct the original plaintext.
Mathematically, the process can be illustrated as follows:
Step I: RSA Attack:
(1)
Step II: Hill Attack:
(From equation 1)
(2)
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Step III: Caesar Attack:
(From equation 2)
(3)
Step IV: Transposition Attack:
(From equation 3)
Where:
Ciphertext obtained using the Transposition method
Ciphertext obtained after applying the Caesar cipher
Ciphertexts obtained after applying the Hill cipher
Ciphertexts obtained after applying the RSA algorithm
,
,
,
Attacks on RSA, Hill, Caesar, and Transposition ciphers respectively
Generating RRS Sequence
To generate a Sequence of Reduce Residue System
a modulus is selected such that the number of
elements in the set corresponds to
, where represents the total number of ASCII characters,
and
denotes Euler’s Totient Function.
Let
Then:
Thus, the RRS sequence modulo 255 can be represented as:
Assigning this generated RRS sequence
to the corresponding ASCII character values allows the
replacement of standard ASCII encoding with residue-based encoding. This substitution increases encryption
complexity, as each ASCII value is represented by its equivalent in the residue sequence, thus enhancing
security before the final RSA encryption phase.
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Table 2: Integrated ASCII Table with RRS
ASCII
Value
Chr
RRS
Value
ASCII
Value
Chr
RRS
Value
ASCII
Value
Chr
RRS
Value
0
NUL
1
43
+
88
86
V
172
1
SOH
2
44
,
89
87
W
173
2
STX
4
45
-
91
88
X
176
3
ETX
7
46
.
92
89
Y
178
4
EOT
8
47
/
94
90
Z
179
5
ENQ
11
48
0
97
91
[
181
6
ACK
13
49
1
98
92
\
182
7
BEL
14
50
2
101
93
]
184
8
BS
16
51
3
103
94
^
188
9
HT
19
52
4
104
95
_
191
10
LF
22
53
5
106
96
`
193
11
VT
23
54
6
107
97
a
194
12
FF
26
55
7
109
98
b
196
13
CR
28
56
8
112
99
c
197
14
SO
29
57
9
113
100
d
199
15
SI
31
58
:
116
101
e
202
16
DLE
32
59
;
118
102
f
203
17
DC1
37
60
<
121
103
g
206
18
DC2
38
61
=
122
104
h
208
19
DC3
41
62
>
124
105
i
209
20
DC4
43
63
?
127
106
j
211
21
NAK
44
64
@
128
107
k
212
22
SYN
46
65
A
131
108
l
214
23
ETB
47
66
B
133
109
m
217
24
CAN
49
67
C
134
110
n
218
25
EM
52
68
D
137
111
o
223
26
SUB
53
69
E
139
112
p
224
27
ESC
56
70
F
142
113
q
226
28
FS
58
71
G
143
114
r
227
29
GS
59
72
H
146
115
s
229
30
RS
61
73
I
148
116
t
232
31
US
62
74
J
149
117
u
233
32
64
75
K
151
118
v
236
33
!
67
76
L
152
119
w
239
34
"
71
77
M
154
120
x
241
35
#
73
78
N
157
121
y
242
36
$
74
79
O
158
122
z
244
37
%
76
80
P
161
123
{
247
38
&
77
81
Q
163
124
|
248
39
'
79
82
R
164
125
}
251
40
(
82
83
S
166
126
~
253
41
)
83
84
T
167
127
DEL
254
42
*
86
85
U
169
Improved Composed Hybrid
The improved composed hybrid encryption method involves a two-stage processes. Firstly, the generated
sequence of Reduced Residue System (RRS) will be applied to replace the characters of the ciphertexts
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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obtained after third encryption (Hill cipher encryption,
) with their corresponding assigned numerical values
as in Table 2. In the second stage, the resulting numerical string is encrypted using RSA encryption techniques.
Procedure for Encryption /Decryption Process of the Modified Hybrid
Encryption Process:
1. Select a transposition key
2. Rearrange the plaintext characters base on the key
to have ciphertexts
as:
that is
(4)
Where is the plaintext,
, and is the number of characters in the plaintext.
3. Select Caesar key
4. Compute
, to obtain a new ciphertexts as:
that is
(5)
where
.
5. Split equation (4.5) as column vectors:
(6)
6. Replace the column vectors of (4.6) with their corresponding ASCII values to have:
(7)
7. Choose an invertible key matrix
8. Compute
, to have:
that is
(8)
9. Convert the values of equation (4.8) to their corresponding characters in ASCII to obtain
(9)
10. Select
such that
, that is value greater than or equals to total number of characters in
ASCII.
11. Generate sequence of Reduce Residue System then, assign each character of the ASCII with a
sequential value of RRS, denoted as:
that is
(10)
12. Replace each letter of equation (4.9) with the corresponding value of
, to have:
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that is
(11)
13. Select RSA public key such that GCD
, where is the encryption key, is the RSA
modulus and
is the Euler number.
14. Encrypt equation (4.11) using RSA method, with a public key , to obtain a final ciphertexts
as:
(12)
Thus, the improved composed cipher becomes:
Decryption Process
The decryption process involves reversing the above encryption process:
1. Compute RSA private key using RSA key equation
2. Decrypt the ciphertexts in equation (4.12) to recover
as in equation (11):
3. Map each value of
with the corresponding value in RRS
then, replace it with corresponding values in
ASCII to recover
as:
(as in equation (9))
where
and ASCII characters.
4. Convert the characters in
to numerals corresponding to ASCII characters to recover
which is the Hill
ciphertexts, as:
(as in (8))
5. Decrypt the resulting ciphertexts
using inverse key matrix to recover
:
to have:
(as in equation (7))
6. Convert the values of
to corresponding characters in ASCII to recover
:
(as in equation (6))
7. Shift each character of
back to a fixed number of positions upward the ASCII based on the Caesar key
to recover
as:
.
(as in equation (4))
8. Rearrange the letters of
back to the original order using the transposition key
to recover the plaintexts
as:
that is:
Thus, the combined decryption process becomes:
Where:
RSA decryption function
Hill decryption function
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Caesar decryption function
Transposition decryption function
Inverse of Hill cipher key matrix
ASCII values of the Hill ciphertexts
RRS value of the ciphertexts
is the ciphertexts of the modified hybrid.
RSA secret key RSA modulus
Caesar key
Application of the Improved Hybrid Cryptosystem
Suppose, to encrypt the message “The message is confidential. Please, keep it secret.”
Firstly, using Transposition to encrypt the statement with the transposition key
, we have:
Table 3: Encryption Table for Transposition Cipher
The ciphertexts consists of the characters read from the top left box going down the column to have ciphertexts
as: Tsifil irhssiaekteea dlae t gce.ses.meon epee ntP, c
Secondly, encrypt the ciphertexts
using Caesar method with key,
to obtain
as:
, illustrated below:
(From ASCII Table)
To have a ciphertexts
as: Wvlilo#lukvvldhnwhhd#godh#w#jfh1vhv1phrq#hshh#qwS/#f
Thirdly, encrypt ciphertexts
using Hill cipher technique with encryption matrix key
as:
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To obtain a new ciphertexts
as:
03106711004712206500705703111605802800502710003403512108000412504211505307806502504209402
4106107030114048077000072028113127045015093078065035124125082123039
USCn/zABEL9USt:FSENQESCd”#yPEOT}*s5NAEM*^CANjkRSr0MNULHFSqDEL-SI]NA#|}R{‘
Fourthly, apply Sequence of Reduce Residue System (RRS) with , to obtain
as:
, to obtain
as:
06213421809424413101411306223211605801105619907107324216100825108622910615713105208618804
9211212061227043197001146058226254091031184157131073248251218247079
Next, generate RSA Public key
and Private key
as:
Let and
with then compute the decryption exponent , such that
, with ,
to have .
Thus, Public key
is , Private key
is
Table 4: Converted ciphertexts
to RRS
ASCII
Value
031
067
110
047
122
065
007
057
031
116
058
028
005
Chr
A
BEL
t
FS
RRS
Value
062
134
218
094
244
131
014
113
062
232
116
058
011
ASCII
Value
027
100
034
035
121
080
004
125
042
115
053
078
065
Chr
RRS
Value
056
199
071
073
242
161
008
251
086
229
106
157
131
ASCII
Value
025
042
094
024
102
101
030
114
020
099
000
072
028
Chr
j
K
RRS
Value
052
086
188
049
211
212
061
227
043
197
001
146
058
ASCII
Value
113
127
045
015
093
078
065
035
124
101
110
123
039
Chr
R
RRS
Value
226
254
091
031
184
157
131
073
248
251
218
247
079
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From Table 4, the new ciphertexts becomes:
06213421809424413101411306223211605801105619907107324216100825108622910615713105208618804
9211212061227043197001146058226254091031184157131073248251218247079
Finally, encrypt ciphertexts
using the public key to obtain final ciphertexts
as:
that is
To have final ciphertexts:
1563143290360650412953091561053461311221291262661462002532193782733213872680412562733
07025284270379320274124001311131061254252380023268041146328378329342037
Decryption process
Reverse the process of the above encryption process to recover the original message.
SUMMARY AND CONCLUSION
This study introduces an improved hybrid cryptosystem that inserts a Reduced Residue System (RRS)
substitution between Hill and RSA. The implemented pipeline is Transposition → Caesar → Hill → RRS →
RSA (baseline omits RRS). Using modulus M = 255, Hill outputs (mapped to ASCII indices) are replaced by
residues from a secret RRS mapping, severing direct numeric correlation with ASCII before RSA encryption.
Even if RSA is compromised, an adversary recovers only RRS-coded residues, forcing a combinatorial
mapping reconstruction rather than direct plaintext recovery. The scheme was tested by encrypting and
decrypting a short message; empirical results show the RRS layer increases attacker workload and enhances
key-generation/management, security, performance, and overall efficiency—producing a more robust hybrid
encryption solution.
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