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A Cryptanalytic Approach to Breaking RSA Encryption Using Public
Key, with Proposed Improvement Leveraging Prime Number
Distribution Patterns
Abubakar T. U.
1
, Ibrahim A. A.
2
, Garba A. I.
2
, Abubakar S. F.
3
, James T. O.
3
, Sarki M. N.
3
, Mua’azu S.
B.
3
, Shehu S.
4
, and Muhammad A. H.
5
1
Department of Mathematics, ShehuShagari College of Education, Sokoto., 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 Science, Mathematics Unit, State Collage of Basic & Remedial Studies
DOI: https://dx.doi.org/10.51244/IJRSI.2025.1210000338
Received: 10 November 2025; Accepted: 16 November 2025; Published: 22 November 2025
ABSTRACT
RSA algorithm, a widely used public-key cryptosystem, relies on the difficulty of factoring large composite
numbers into their prime factors. However, advancements in computational power and factorization techniques
have introduced potential threats to its security. As RSA remains a cornerstone of modern cryptography, the
need for improved in its security measures is paramount in the face of evolving computational challenges. This
study presents a cryptanalytic examination of the RSA encryption scheme and proposes an improvement that
leverages prime number distribution patterns to strengthen data security. Several mathematical methods which
involve RSA key generation and its encryption/decryption process, ASCII Table, mapping as well as the Sieve
of Eratosthenes were applied in the study. The research analyses how RSA public parameters and encoding
methods can reveal structural weaknesses when subjected to mathematical scrutiny. To address these
vulnerabilities, a modified scheme is introduced, in which plaintext characters are mapped using prime
distribution patterns. This substitution increases ciphertexts randomness and minimizes predictable patterns
between plaintexts and ciphertexts. Experimental evaluation demonstrates that the proposed improvement
enhances resistance to analytical attacks while maintaining RSA operational compatibility. The study
contributes to the on-going development of more secure and efficient public-key cryptographic systems.
Keywords: RSA, Encryption, Decryption, Cryptanalysis, Mapping and Prime numbers.
INTRODUCTION
The advent of public-key cryptography revolutionized data security by introducing a mechanism for secure
communication over untrusted networks. Among various asymmetric cryptographic algorithms, the Rivest–
Shamir–Adleman (RSA) encryption scheme remains one of the most widely adopted due to its mathematical
simplicity and proven resistance to brute-force attacks. Since its introduction in 1977 by Rivest, Shamir, and
Adleman, RSA has been the cornerstone of digital security in applications such as secure email, digital
signatures, and key exchange protocols (Rivest, Shamir and Adleman, 1978). Its security fundamentally relies
on the computational difficulty of factoring a large composite integer , where and are large
prime numbers. The public key in RSA, typically denoted as , is openly distributed, while the private
key remains secret. Theoretically, knowledge of only the public key should not compromise the private key,
(Nitaj, 2016). Nonetheless, recent studies have shown that under specific conditions such as weak key
INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)
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generation, poor randomness in prime selection, or low encryption exponents an attacker can exploit
mathematical patterns to deduce partial information about the private key or factor , (Ariffin & Shehu, 2016;
Wiener, 1990). Cryptanalysis of RSA using the encryption exponent involves exploiting possible
vulnerabilities when the encryption key , modulus , and encoding scheme are public.
This study, therefore, focuses on exploring cryptanalytic techniques that utilize only the public key to test the
boundaries of RSA security. It aims to investigate a cryptanalytic approach to breaking RSA encryption using
public key, with proposed improvement leveraging prime number distribution patterns.
The findings are expected to contribute to both cryptanalysis and cryptographic engineering, providing a
clearer understanding of how RSA can be both broken and improved.
LITERATURE REVIEW
Overview of the RSA Algorithm and Its Mathematical Foundation
The Rivest, Shamir, Adleman (RSA) algorithm, introduced in 1977, remains one of the earliest and most
enduring forms of public key cryptography, (Nadia, Syahril & Sawaluddin, 2020). Its theoretical foundation
lies in modular arithmetic and the difficulty of factoring a large composite number , where and
are prime numbers (Rivest, Shamir & Adleman, 1978). RSA operates on the principle of one way functions:
while it is computationally straightforward to multiply large primes, reversing the process to retrieve the
original factors is considered infeasible with classical computing methods. The security of RSA thus depends
on the hardness of the Integer Factorization Problem (IFP).
Classical Cryptanalytic Techniques against RSA
Early research into RSA vulnerabilities focused primarily on factoring algorithms. The Fermat factorization
method is one of the oldest techniques, effective when the two primes and are close in magnitude
(Aminudina & Cahyonoa, 2021). Subsequently, Pollard’s Rho algorithm improved upon Fermat’s approach
by using pseudo-random number generation to detect nontrivial factors more efficiently.
Kefa (2006), in a review of integer factorization methods, discussed several algorithms including the Elliptic
Curve Algorithm (ECM), Quadratic Sieve (QS), and Number Field Sieve (NFS). Hinek (2009) demonstrated
that it is possible to factor the modulus
if
where
, is a small constant.
Somsuk and Kasemvilas (2013) introduced the Possible Prime Modified Fermat Factorization (PPMFF)
algorithm, which improves the handling of both trivial and non-trivial values.
Overmars and Venkatraman, (2020), in their paper applied mathematical tools and algorithms designed for
factorizing semi-prime numbers using the sum of squares method incorporating mathematical theorems and
principles related to prime factorization and semi-prime numbers to support the development and validation of
the new method.
Aminudin and Cahyono, (2021), provide a practical analysis of Fermat's method alongside Pollard's rho
method, reinforcing the notion that these classical techniques remain vital in the landscape of integer
factorization.
Overmars and Venkatraman, (2021), showed that if a Pythagorean quadruple is known and one of its squares
represents a Pythagorean triple, then the semi-prime is factorized. They proved that to factor a semi-prime, it is
sufficient that only one of these Pythagorean quadruples be known.
Kannan and Mohana, (2023), they delve into the challenges associated with factorization in the context of the
RSA cryptosystem. They also discussed the various programming tools utilized to handle the challenge of
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factoring large numbers and penetrated into the enhancement of the Simple Fermat Primality Test as a means
to expedite the primality testing algorithm.
Despite these advances, even GNFS requires exponential time for sufficiently large RSA moduli (greater than
2048 bits), maintaining RSA practical security under classical computation.
Quantum Threats and Post-Quantum Cryptanalysis
A major turning point in cryptographic research occurred with Shor’s algorithm, which demonstrated that a
quantum computer could factor large integers in polynomial time, effectively breaking RSA’s foundational
assumption, (shor, 1994). Although large-scale quantum computers capable of executing Shor’s algorithm on
RSA-2048 remain theoretical, this revelation has prompted the development of post-quantum cryptography
(PQC).
Researchers are now exploring cryptographic schemes based on lattice problems, multivariate polynomials,
and hash-based systems, which are believed to resist quantum attacks. However, before transitioning away
from RSA entirely, it remains crucial to understand its residual vulnerabilities under classical computation,
particularly through mathematical cryptanalysis using public key inference.
Prime Number Distribution and Cryptographic Implications
Prime numbers form the backbone of RSA encryption. Their distribution, however, is neither entirely random
nor uniform. The Prime Number Theorem (PNT) approximates the number of primes less than a given
integer as
, indicating that primes become sparser as numbers increase (Hardy & Wright, 2008). Despite
their apparent irregularity, primes exhibit deep mathematical structures that can be modelled through
probabilistic and analytical techniques.
Recent studies (Ameha, 2021) have explored formulas for predicting prime occurrences, revealing patterns in
prime gaps, residue sequences, and modular symmetries. Such findings have two major implications:
1. Cryptanalysts may use these patterns to narrow down prime search spaces, thus improving
factorization attempts.
2. Cryptographers may exploit these same distributions to generate primes more securely, by
introducing pseudo-random distortions that obscure detectable regularities.
Drawing from these insights, the present research proposes a cryptanalytic approach that leverages the
distribution patterns of prime numbers to infer potential weaknesses in RSA when the public key is
available. The paper introduces a new cryptographic improvement embedding prime-distribution-based
substitution of ASCII characters prior to encryption. This modification aims to enhance security by
obfuscating plaintext patterns before they enter the RSA process, effectively creating a layered defence that
combines number-theoretic complexity with symbolic transformation. In essence, this study extends the
scope of RSA research beyond normal factorization-based cryptanalysis to a pattern-oriented and
distribution-aware approach, offering a new lens through which RSA mathematical foundation can be both
examined and strengthened.
METHODOLOGY
This study employed several mathematical methods which involve RSA key generation and its
encryption/decryption process, ASCII Table, mapping as well as the Sieve of Eratosthenes.
ASCII Table
ASCII (American Standard Code for Information Interchange) is a character encoding standard that assigns
numeric values to characters and symbols. The complete 7-bit ASCII table and corresponding decimal
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equivalents are presented in the table below:
Table 1:
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
RSA Encryption/Decryption Process
Key Generation:
RSA key generation involves creating a public key (used for encryption) and a private key (used for
decryption).
Step I: Choose two distinct primes with .
Step II: Compute and
Step III: Choose a public key such that GCD
.
Step IV: Compute the private key such that
.
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Encryption Process
Compute the ciphertexts using the public key :
Decryption Process
Decrypt the ciphertexts using the private key :
Concept of Mapping
Mapping refers to a fundamental concept that describes the relationship between two sets, where each
element in one set is associated with one or more elements in another set. Formally, if and are sets, a
mapping or function from to is defined as:
such that for every element , there exists a unique element satisfying . Here, is called
the domain, is the codomain, and the set of all actual images in is the range of the function (Hardy
& Wright, 2008).
Sieve of Eratosthenes Algorithm
The Sieve of Eratosthenes is a classical algorithm for generating prime numbers up to a given limit , (Ameha,
2021).
Algorithm Steps:
1. Create a list of numbers from to .
2. Begin with the first prime .
3. Mark all multiples of each prime as non-prime.
4. Continue until all numbers are processed.
5. The unmarked numbers are primes.
RESULT AND ANALYSIS
This section presents the results and analysis of the research. The findings demonstrated that RSA is vulnerable
to attacks that exploit the public key exponent, and that patterns of prime numbers can be used to enhance the
efficiency and security of the RSA algorithm.
Cryptanalysis of RSA Using Encryption Key
Cryptanalysis of RSA using the encryption exponent involves exploiting vulnerabilities when the encryption
key , modulus , and encoding scheme are public. In such cases, an attacker may recover plaintext by
systematically encrypting all possible character codes and comparing them with the ciphertexts. Below is an
illustration of the cryptanalysis process:
Let be a public key
Set of standard codes comprising letters, numerals and symbols.
Plaintext, where
Ciphertexts
, where
Attack
, where
, number of elements in
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To have .
Mapping the values in with the corresponding values in , the original message will be recovered.
Suppose a plaintext message “Mission completed” was encrypted using RSA public key
,
and each character of the message was assigned with its corresponding value in ASCII as:
Plaintext
Encryption
M
i
d
To have encrypted message as:
An attacker can recover the original message by encrypting all the characters in the ASCII table using the
public key
to have encrypted ASCII values as:
NUL
SOH
DEL
Mapping the values in the ciphertexts with the corresponding values in , the original message
“Mission completed” will be recovered.
Proposed Improvement of the RSA Algorithm Leveraging Prime Number Distribution Patterns.
Applying patterns of prime numbers to improve RSA algorithm involves four stages. Firstly, generate pattern
of prime numbers; secondly, replace the characters of ASCII with the generated pattern sequentially. Thirdly,
express the given plaintext in terms of assigned prime numbers and lastly, encrypt the resulting values using
RSA method of encryption.
Encryption Process of the Improved Algorithm
Below is the step by step of the encryption process:
Step I: Generate pattern of prime numbers of specific range.
Step II: Map each value of the generated prime to a corresponding value in ASCII
Step III: Express the given plaintext in terms of assigned prime numbers as
Step IV: Generate RSA public key and private key .
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Step V: Encrypt
using the RSA public key to obtain ciphertexts
Decryption Process of the Improved Algorithm
To decrypt the generated ciphertexts the process of encryption is reverse using RSA decryption exponent ,
as in the steps below:
Step I: Decrypt the ciphertexts to obtain
as
Step II: Express the values of
to the corresponding values in ASCII
Step III: Convert the values to the corresponding characters to obtain the plaintext .
Pattern of Prime Numbers using Sieve of Eratosthenes Technique
The Sieve of Eratosthenes is an ancient algorithm used to find all prime numbers up to a given limit. By
applying this technique, pattern of prime numbers can be generated sequentially. For instance, to generate
prime numbers corresponding to the total number of characters in ASCII, ranging from 31 to 800 (that is
, , where
is prime). This is illustrated in table 2:
Table 2: Sieve of Eratosthenes for Prime Numbers
,
31
33
37
39
411
413
417
419
41
43
47
49
421
423
427
429
51
53
57
59
431
433
437
439
61
63
67
69
441
443
447
449
71
73
77
79
451
453
457
459
81
83
87
89
461
463
467
469
91
93
97
99
471
473
477
479
101
103
107
109
481
483
487
489
111
113
117
119
491
493
497
499
121
123
127
129
501
503
507
509
131
133
137
139
511
513
517
519
141
143
147
149
521
523
527
529
151
153
157
159
531
533
537
539
161
163
167
169
541
543
547
549
171
173
177
179
551
553
557
559
181
183
187
189
561
563
567
569
191
193
197
199
571
573
577
579
201
203
207
209
581
583
587
589
211
213
217
219
591
593
597
599
221
223
227
229
601
603
607
609
231
233
237
239
611
613
617
619
241
243
247
249
621
623
627
629
251
253
257
259
631
633
637
639
261
263
267
269
641
643
647
649
271
273
277
279
651
653
657
659
281
283
287
289
661
663
667
669
291
293
297
299
671
673
677
679
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301
303
307
309
681
683
687
689
311
313
317
319
691
693
697
699
321
323
327
329
701
703
707
709
331
333
337
339
711
713
717
719
341
343
347
349
721
723
727
729
351
353
357
359
731
733
737
739
361
363
367
369
741
743
747
749
371
373
377
379
751
753
757
759
381
383
387
389
761
763
767
769
391
393
397
399
771
773
777
779
401
403
407
409
781
783
787
789
Where multiples of two and that of five are deleted, hence, the required pattern of primes is:
31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151,
157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271,
277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409,
419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557,
563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683,
691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787.
Applying Patterns of Prime Numbers to Improve the Security of RSA Algorithm
The generated pattern of primes will be applied to replace letters and characters of a plaintexts derived from
ASCII table. Below is the integrated ASCII table with assigned prime numbers:
Table 3: Integrated ASCII Table
Dec
0
1
2
3
4
5
6
7
8
9
Chr
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL
BS
HT
31
37
41
43
47
53
59
61
67
71
Dec
10
11
12
13
14
15
16
17
18
19
Chr
LF
BT
FF
CR
SO
SI
DEL
DC1
DC2
DC3
73
79
83
89
97
101
103
107
109
113
Dec
20
21
22
23
24
25
26
27
28
29
Chr
DC4
NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
127
131
137
139
149
151
157
163
167
173
Dec
30
31
32
33
34
35
36
37
38
39
Chr
RS
US
SPC
!
“
#
$
%
&
‘
179
181
191
193
197
199
211
223
227
229
Dec
40
41
42
43
44
45
46
47
48
49
Chr
(
)
*
+
,
-
.
/
0
1
233
239
241
251
257
263
269
271
277
281
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Dec
50
51
52
53
54
55
56
57
58
59
Chr
2
3
4
5
6
7
8
9
:
;
283
293
307
311
313
317
331
337
347
349
Dec
60
61
62
63
64
65
66
67
68
69
Chr
<
=
>
?
@
A
B
C
D
E
353
359
367
373
379
383
389
397
401
409
Dec
70
71
72
73
74
75
76
77
78
79
Chr
F
G
H
I
J
K
L
M
N
O
419
421
431
433
439
443
449
457
461
463
Dec
80
81
82
83
84
85
86
87
88
89
Chr
P
Q
R
S
T
U
V
W
X
Y
467
479
487
491
499
503
509
521
523
541
Dec
90
91
92
93
94
95
96
97
98
99
Chr
Z
[
\
]
^
_
`
a
b
c
547
557
563
569
571
577
587
593
599
601
Dec
100
101
102
103
104
105
106
107
108
109
Chr
d
e
f
g
H
i
j
k
l
m
607
613
617
619
631
641
643
647
653
659
Dec
110
111
112
113
114
115
116
117
118
119
Chr
n
o
p
q
R
s
t
u
v
w
661
673
677
683
691
701
709
719
727
733
Dec
120
121
122
123
124
125
126
127
Chr
x
y
z
{
|
}
~
DEL
739
743
751
757
761
769
773
787
Utilizing the integrated ASCII table to improve the security of the message encrypted in section 4.1, “Mission
completed” using RSA public key
:
Plaintext
Encryption
M
i
d
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To have new encrypted message as:
The above ciphertexts will be difficult to attack using public key as in normal RSA. With this improvement the
original message can be recovered using decryption key as illustrated below:
Decryption Process:
Computing the decryption exponent , such that
, with ,
to have
.
To decrypt the ciphertexts using
and taking its corresponding value character from Table
3, as:
ASCII Value Plaintext
M
i
d
Hence, the plaintext message “Mission completed” has been obtained.
SUMMARY AND CONCLUSION
The proposed approach was implemented by assigning prime numbers to each ASCII character, covering the
full range of values (0 – 127). Instead of directly encrypting ASCII codes, each character was mapped to a
unique prime number within a defined range (31–800). The mapped primes were then encrypted using the RSA
scheme with modulus and exponent. Every ASCII character had a distinct prime representative. This mapping
introduced an additional layer of substitution before RSA encryption, increasing ciphertexts diversity.
Ciphertexts generated from prime values were larger than those generated directly from ASCII codes. The use
of primes patterns avoided direct correlation between ciphertexts values and ASCII frequencies. Frequency
analysis, which exploits repeated characters in plaintext, became significantly harder.
The results also, highlighted that the integration of prime pattern mapping with RSA improves security in two
major ways: normal RSA encrypts small integers (ASCII 0–127), which can be predictable and by replacing
ASCII values with pattern of primes, the plaintext space becomes unpredictable. Furthermore, the finding
showed vulnerability of RSA through public key-based.
Conclusively, the proposed improved RSA algorithm increases resistance against frequency analysis,
ciphertexts pattern recognition, and direct recovery from public key attacks.
REFERENCES
1. Ameha, T. T. (2021). Advanced mathematical formulas to calculate prime numbers. Strategic
Planning, Commercial Bank of Ethiopia, Addis Ababa, Ethiopia.
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