BIT112 Mathematics for IT: Public Key Cryptography and RSA Encryption

This report explores the concepts of encryption, focusing on the RSA encryption system and its application in securing communications. It defines encryption, highlighting its role in protecting data against cyber threats. The report outlines the objectives, which include defining encryption, understanding its significance, and reviewing RSA's working process. The RSA encryption system, a crucial topic, is described, detailing its asymmetric nature and the use of public and private keys. The report explains the key generation process, including prime number selection and key derivation, and provides an example of encryption and decryption. Various methods used in encryption, such as public key cryptography, Wolfram Alpha, and factorization, are explained. The report concludes by summarizing the importance of encryption, emphasizing RSA's role in data protection, and public-key cryptography for encryption. The report is based on the assignment brief which requires the student to generate their own private and public keys, encrypt their student ID, and attempt to break the private key using the Wiener attack. The student also needs to show the process using Wolfram Alpha.

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Mathematics for IT

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Table of Contents
Introduction.................................................................................................................................................2
Objectives....................................................................................................................................................2
RSA encryption system................................................................................................................................2
Working of RSA encryption system.............................................................................................................4
Methods......................................................................................................................................................5
Public key cryptography..........................................................................................................................6
Wolfram Alpha........................................................................................................................................6
Factorization method..............................................................................................................................7
RSA encryption........................................................................................................................................7
Conclusion...................................................................................................................................................7
References...................................................................................................................................................9

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Introduction
Encryption is a kind of process used in the communication systems for encoding a signal in
a way that only authorized parties may access with the help of private keys. It is an
effective technique that is now used in the security approaches for protecting data against
cyber-crimes and attacks. Dhakar, Gupta, and Sharma, (2012) reported encryption is the
technique by which data is converted into personal code which is capable of hides the
information of consumers in a reliable manner. This report mainly focuses on the various
encryption techniques used in the communication systems and review the working process
of RSA encryption methodology. RSA is an advanced encryption system that has the
potential to address unauthorized access from the systems and enhance the privacy of data
effectively. This report will cover various parts including RSA encryption, working of RSA,
methods used in encryption and conclusion.
Objectives
There are the following objectives of the research report:
 To define the concept of the encryption process
 To understand the significance of the encryption system
 To review the working of the RSA encryption system
 To determine the methods used in the encryption
RSA encryption system
It is defined as Rivest Shamir Adleman which is an effective cryptosystem utilized in the
communication networks for protecting the transmitted signals. The key feature of RSA
encryption is that it is capable to encrypt and decrypt larger signals in less time which
leads effectiveness of the system. In this field of RSA, the encryption key is public and it is
not similar to the decryption key. Mahalle and Shahade, (2014) determined that in the case
of the RSA encryption system, asymmetry is grounded on the practical difficulty of the
factorization of two larger numbers.

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It is examined that RSA is an asymmetric cryptography process that means it works on two
encryption keys including private and public. Private Key is used for performing personal
data communication while the public key is given to every consumer in order to access
transmitted data. The key idea behind the RSA encryption technique is grounded on the
fact that it is complex to factorize a huge integer. The public key includes two numbers in
which one is the product of two prime numbers. Khanezaei and Hanapi, (2014) proposed a
research study and determined that in the case of the RSA encryption system, signals are
encrypted with a code defined as a public key that may be transferred to everyone.
Because of some distinct mathematical properties of the RSA encryption, once a signal has
been converted using a public key, it may be decrypted by another key easily. Every RSA
consumer has a key pair including private and public keys in order to manage the privacy
of data and access the transmitted signals easily. From a recent investigation, it is
determined that the concept behind RSA encryption may be utilized in combination with
other encryption systems that may prove the authenticity and integrity of a signal
(Shankar, 2018). It is not mainly utilized to encrypt the complete transmitted signal as it is
less effective and not able to protect data against malicious programs. In order to make a
more effective process, a data file will be encrypted with a symmetric key and then the
developed key will be encrypted using the RSA encryption system.
In this approach, only an entity that has access to the RSA key will be capable to convert the
transmitted signal into the original information. Mahajan and Sachdeva, (2013) observed
that the RSA encryption system may be utilized in various communication systems
including OpenSSL, cryptlib, wolfCrypt, and other cryptographic libraries. It is found that
there are major two applications of RSA encryption including TLS and PGP encryption.
Moreover, it is utilized to make a private connection between VPN consumers and VPN
servers. Under network protocols such as OpenVPN, the TLS system may utilize the
working of RSA encryption in order to exchange keys and develop a secured network. It is
determined that RSA encryption is completely grounded on the principle which it is simple
to multiply huge numbers.

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Due to the larger difficulty in breaking RSA, it is generally utilized where encryption is
needed. Liu, et al., (2018) demonstrated that RSA can be utilized to ensure web servers are
secured since only the actual servers would have their own personal key. It is capable to
control and address man-in-the-middle attacks where the criminals intercept an
interconnection and indicate personal information (Singh, 2013). It is examined that RSA
encryption is no longer a secured system rather than other methods but it is beneficial for
encrypting the transmitted signal. One major attack on the RSA system bypass the
algorithm and a computing network may effectively compute the largest divisor of two
prime numbers with the help of the Euclidean algorithm.
Working of RSA encryption system
It is observed that the RSA process utilizes prime numbers which are larger in magnitude.
There are numerous factors included in the case of RSA encryption such as trapdoor
functions, Carmichael's totient and so on. Ayele and Sreenivasarao, (2013) observed that
RSA encryption works under the premise that the algorithm is simple to compute in one
direction but it is not possible to perform the reverse process. There are two parts of the
RSA working such as generating a public key and generating the private key.
Generating public key
There are following steps involves while generating the public key in the RSA algorithm
which are reviewed with an example:
 First chose two prime numbers, for example, p= 53 and q= 59
 Now, the first part of the public key is determined by multiplying prime numbers:
 N= p*q=3127
 We also require a small exponent that is “e” which should be an integer and not a
factor of “n”
 Assumed the value of n is 3
 Finally, the public key is determined is then determined by taking mode of “n” and
“e”.
Generating private key

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For finding private key following steps could be utilized:
 Consumers require to determine (n) which is given below:Φ
 (n) = (P-1)(Q-1)Φ
 So, (n) = 3016Φ
 Now, private key (d) may be obtained using the below formula:
 d = (k* (n) + 1) / eΦ
 Here, k= 2
 Therefore, private key= 2011
Now, it is observed that the public key is 3127 and the private key is 2011 which will be
helpful for encrypting the signal.
Example
We need to encrypt “HI” in an encrypted signal using the RSA algorithm.
Here, H= 8 and I= 9
Encrypting signal:
So, encrypted signal c= 89e mod n
Therefore, encrypted data is 1394
Decrypting signal:
Decrypted data= cd mod n
So, decrypted data is 89 that represents "HI” as a transmitted signal.
Methods
There are numerous kinds of methods used in the encryption process for converting the
transmitted signals into a unique code which are described below:

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Public key cryptography
It is defined as asymmetric cryptography which utilizes pairs of keys including private and
public keys in order to convert the transmitted signals. Unlike symmetric key process that
relies on one key for performing encryption and decryption, each key perform a particular
function. It is argued that the public key is capable to encrypt the signal while the private
key is adopted for decrypting the shared information. There are major two applications of
public-key cryptography, for example, digital signatures, and encryption (Meneses, et al.,
2016). In the case of digital signature, content is signed with a private key used by the
consumer and verified with the help of the public key. In the encryption process, public-key
cryptography is used to encrypt the transferred signals with the help of the public key and
the private key is explored for decrypting signals. There are numerous key features of
public-key cryptography which are highlighted below:
 Various keys are adopted for performing encryption and decryption
 Every receiver possesses a specific decryption key which is called a private key
 The consumer requires providing an encryption key for converting the transferred
signals that called public key
 It is found that the encryption algorithm is effective and appropriate for prohibiting
criminals
 There are three kinds of public-key cryptography systems involve in the
communication channels such as RSA cryptosystem, ElGamal cryptosystem, and
Elliptic Curve cryptosystem.
Wolfram Alpha
It is a kind of computational search engine which is mainly used for providing computing
answers and knowledge in the relation of communication channels. With the help of this
platform, consumers can perform analysis and produce research reports. In the case of
communication systems, Wolfram Alpha is utilized for calculating modulus and totient
which are helpful for encrypting and decrypting the transferred signals (NaQi, et al., 2013).
Using the operations of Wolfram Alpha consumers can easily find the modulus of any
number which reduces complexity and enhance effectiveness. In which users need to put

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two values for finding modulus and developed programs automatically find out the answer
and show on the computer screen.
Factorization method
It is a kind of method consisting of writing a number as a product of numerous factors
including smaller objects. In the case of the encryption system, the factorization method is
utilized for breaking the developed keys. With the help of this process, the consumers can
check the security of the developed encryption system and convert a signal into a number
of signals.
RSA encryption
It is defined as an algorithm that is generally utilized by the consumers for encrypting and
decrypting the signals in order to secure sensitive information. It is part of the asymmetric
cryptographic technique that contains two types of keys for performing operations, for
example, private and public keys. It is also known as public-key cryptography as one of the
keys may be provided to anyone.
Conclusion
From the above evaluation, it may be concluded that encryption is the best technique for
securing data from criminals and cyber-attacks. This research report provided complete
information in the case of encryption and cryptography systems. There are two kinds of
techniques used in encryption such as the RSA encryption system and public-key
cryptography. It is found that RSA encryption includes two keys such as private and public
keys where the public key is used for encrypting the signal and the private key is used for
decrypting the transmitted signal. This proposed research provided depth information
about the working of the RSA encryption system and evaluated with an example for a
better understanding encryption process. It is found that in the case of public-key
cryptography, the encryption algorithm is complex enough for prohibiting criminals from
deducing the plaintext from the ciphertext and the encryption key. Therefore, it is reported
that the utilization of RSA encryption is beneficial for protecting data against cyber-attacks
and public-key cryptography can be used for encrypting the information.

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References
Ayele, A.A. and Sreenivasarao, V., (20130 A modified RSA encryption technique based on
multiple public keys. International Journal of Innovative Research in Computer and
Communication Engineering, 1(4), pp.859-864.
Dhakar, R.S., Gupta, A.K. and Sharma, P., (2012) Modified RSA encryption algorithm
(MREA). In 2012 Second International Conference on Advanced Computing & Communication
Technologies, 12(6), pp. 426-429.
Khanezaei, N. and Hanapi, Z.M., (2014) A framework based on RSA and AES encryption
algorithms for cloud computing services. In 2014 IEEE Conference on Systems, Process, and
Control (ICSPC 2014), 12(8), pp. 58-62.
Liu, Y., Tang, S., Liu, R., Zhang, L. and Ma, Z., (2018) Secure and robust digital image
watermarking scheme using logistic and RSA encryption. Expert Systems with
Applications, 97(7), pp.95-105.
Mahajan, P. and Sachdeva, A., (2013) A study of encryption algorithms AES, DES, and RSA
for security. Global Journal of Computer Science and Technology, 12(6), pp. 12-18.
Mahalle, V.S. and Shahade, A.K., (2014) Enhancing the data security in Cloud by
implementing hybrid (Rsa & Aes) encryption algorithm. In 2014 International Conference
on Power, Automation and Communication (INPAC), 12(7), pp. 146-149.
Meneses, F., Fuertes, W., Sancho, J., Salvador, S., Flores, D., Aules, H., Castro, F., Torres, J.,
Miranda, A. and Nuela, D., (2016) RSA encryption algorithm optimization to improve
performance and security level of network messages. IJCSNS, 16(8), p.55.
NaQi, W.W., Zhang, J., Wang, W., Zhao, J., Li, J., Shen, P., Yin, X., Xiao, X. and Hu, J., (2013)
Analysis and Research of the RSA Algorithm. Asian network for scientific Information, 12(9),
pp.1818-1824.
Shankar, K., (2018) An optimal RSA encryption algorithm for secret images. International
Journal of Pure and Applied Mathematics, 118(20), pp.2491-2500.

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Singh, G., (2013) A study of encryption algorithms (RSA, DES, 3DES, and AES) for
information security. International Journal of Computer Applications, 67(19), pp. 12-18.