A Note on Public Key Cryptosystems

Information is a valuable asset and must be kept confidential, have integrity and be available in order to be worth its name and be credible. Therefore in a bid to ensure this, strong mathematical algorithms, that involve the use of a pair of keys which are different from one another (public and private) like the ones that are used by RSA and CISCO to mention but a few, are employed and used in the processes of data encryption and decryption. This system of using two keys, one public and the other private that are different but mathematically related, to secure information has given rise to the concept of Public Key Cryptography; a concept that attempts to use key management in ensuring the security of information and data against hacker's attacks while both in storage and moving over the internet from one place to another. This paper thus, sets out to analyze the working of the public key cryptography and find ways in which it can be improved to give better information security.

Internet has revolutionized many aspects of our daily lives. Nowadays Internet is used for millions of applications. Many people depend on Internet for several activities like on-line banking, on-line shopping, on-line learning and on-line meetings etc. Huge amount of data travels over the network. Security of data over the network is a critical issue. Making a network secure involves a lot more than just keeping it free from programming errors. Network Security refers to the protection of valuable data against Interception, Interruption, Modification, Fabrication and Non-repudiation. Computer networks are inherently insecure so to protect data over the networks we need some mechanism. Cryptography came into existence to ensure data security. There are various threats to data: Backdoors, denial-of-service attack, direct-access attack, eavesdropping, exploits, indirect-attacks and social engineering and human-error. Cryptography provides protection against security threats. Cryptography means secret writing, the content of original text is scrambled to produce coded text and job of intruders becomes difficult. Secret-writing is the strongest tool of cryptography which protects the data. Cryptography is used to ensure confidentiality, integrity and availability of data by using private and public key cryptography algorithms. Private and Public key algorithms are used to transform original (readable) messages into unreadable jumbles. This paper describes the comparison of Private (symmetric) and Public (asymmetric) key algorithms.

This paper shows four models of Public Key Cryptography (PKC) that doesn't use or simplifies the use of Public Key Infrastructure (PKI): ID-Based, Self-Certified, Certificateless and Certificate-Based. These models are showing on operations and properties. At the end there is a comparative table that summarizes the main characteristics.

10th Anniversary Edition, 2009

Ideas from computational complexity play a very fundamental role in cryptography. A formal prerequisite for this module is familiarity with basic concepts of algorithm analysis and computational complexity, including the big-O notation. Students are also assumed to be able to do basic programming in a language such as C, C++ or Java. Necessary material from elementary number theory is covered in the module, though familiarity with modular arithmetic and some very basic logic (such as proof by contradiction) will be useful. Since mostly number theoretical algorithms are discussed in this module, it is very important to emphasize that if the input to such an algorithm is an integer , n then the size of the input is taken to be 2 log ( ) 1, n    number of bits needed to store , n not the magnitude of . n This is a point that is easy for students to forget, therefore it is worth repeating. While providing necessary background in elementary number theory, students will be given a gentle introduction to some fundamental notions in abstract algebra such as groups, rings, and fields. Students who already have this background may skip that part. (Even if they saw the material before, a quick review would not hurt). If the mathematical maturity of the students is not at the right level, some of the proofs could be optional. The software used in this module is Maple 12, though it should also work with some of the earlier versions of Maple. There are several projects to choose from at the end. Given the nature of the subject some of these projects are research oriented and not completely prescribed. Others are of hands-on type and also not completely prescribed, some of the details need to be determined by the instructor. The exercises inserted throughout the module and the projects at the end can be used as an assessment tool.

Cryptography is the technique of transforming an intelligible message into unintelligible format so that the message can't be read or understood by an unauthorized person during its transmission over the public networks. A number of cryptographic techniques have been developed over the centuries. With technological advancement, new techniques have been evolved significantly. Public key cryptography offers a great security for transmitting data over the public networks such as Internet. The popular public key cryptosystems like RSA and Diffie-Hellman are becoming slowly disappearing because of requirement of large number of bits in the encryption and decryption keys. Elliptic Curve Cryptograph (ECC) is emerging as an alternative to the existing public key cryptosystems. This paper describes the idea of Elliptic Curve Cryptography (ECC) and its implementation through two dimensional (2D) geometry for data encryption and decryption. This paper discusses the implementation of ECC over prime field. Much attention has been given on the mathematics of elliptic curves starting from their derivations.

2016

Elliptic curve cryptography (ECC) is an approach to public key Cryptography based on the algebraic structure of Elliptic curves over finite field. Elliptic curves are also used in several integer factorization algorithms that have applications in cryptography. An elliptic curve in cryptography was suggested independently by Neal Koblitz and Victor S. Millar in 1985. Elliptic curve cryptography algorithms entered wide use in 2004 to 2005.Cryptography comes from Greek words meaning “hidden writing”. Cryptography converts readable data or clear text into encoded data called cipher text. By definition cryptography is the science of hiding information so that unauthorized users cannot read it. It involves Encryption and decryption of messages. Encryption is the process of converting a Plain text into cipher text and decryption is the process of getting back the original Message from the encrypted text. ECC is a newer approach to public-key cryptography based on the algebraic structure of...

Network is a collection of interconnected nodes which are spread over a large region. A node can be any device such as personal computer, mobile phone, tablet, WAP devices, pager, etc. Data is transmitted over channel of this network, which is prone to security threats such as loss of confidentiality, loss of integrity, fabrication attacks, etc. So, there is a need to secure this data transmission. It is achieved through Cryptography [9].The present work focuses on cryptography to secure the data while transmitting in the network. Firstly, the data which is to be transmitted from sender to receiver in the network must be encrypted using the encryption algorithm in cryptography. Secondly, by using decryption technique the receiver can view the original data. Many research papers have been submitted on this cryptographic algorithm.This paper aims at study of the two asymmetric-key algorithms i.e. Rivest-Shamir-Adlemann (RSA) and Elliptic-Curve cryptography (ECC).A comparative analysis of both the algorithms has been done and observed that RSA is one of the effective public key cryptographic algorithms, which needs time and memory whereas ECC provides a strong alternative with smaller key lengths and more secure [1][2].

An asymmetric algorithm is an encryption technique that uses different keys on the process of encryption and decryption. This algorithm uses two keys, public key, and private key. The public key is publicly distributed while the private key is kept confidentially by the user and this key is required at the time of the decryption process. RSA and ElGamal are two algorithms that implement a public key cryptosystem. The strength of this algorithm lies in the bit length used. The degree of difficulty in RSA lies in the factorization of large primes while in ElGamal lies in the calculation of discrete logarithms. After testing, it is proven that RSA performs a faster encryption process than ElGamal. However, ElGamal decryption process is faster than RSA. Both of these algorithms are cryptographic public-key algorithms but have functions in different ways. RSA is a deterministic algorithm while ElGamal is a probabilistic algorithm.

Cryptography techniques play an important role in modern world. The purpose of such techniques is to ensure the contents being unreadable to anyone except for parties who agreed to use some specific scheme. Moreover, current cryptography techniques provide more sophisticated services, such as message integrity, authentication, time stamping, and many others. There are two main approaches for cryptography: private-key cryptography and public-key cryptography (PKC). In this paper we focus on PKC techniques giving a comparison between three main techniques, namely, Public key Infrastructure (PKI), Identity- Based Cryptography (IBC) and Certificate less Public Key Cryptography (CL-PKC). In this research, a brief definition, advantages and disadvantages and analysis of main problem, namely, the revocation problem, are introduced for the three techniques. Also, a variety of available solutions to overcome the revocation problem in each technique are highlighted. Finally, some common applications and schemes for each technique are summarized.

Public key cryptography has become an important means of ensuring confidentiality, notably through its use of key distribution, where users seeking private communication exchange encryption keys. It also features digital signatures which allow users to sign keys to verify their identities. This research presents the innovations in the field of public-key cryptography while also analyzing their shortcomings. We present methods of improving upon the weaknesses that include techniques involving double encryption and mutual authentication. These contributions introduce new levels of security to the subject with ideas to combat man in the middle attacks and other hacker scenarios. Public-key encryption with digital signatures offers both security and data integrity against most attackers.

Algebra for Cryptologists, 2016

In this chapter we describe, at an elementary level, some of the applications of the Group Theory and Number Theory we have developed so far to Cryptology. We emphasise that these "textbook versions" of the applications do not do justice to the complexities that arise in practice, and warn the reader that implementing the mechanisms that we discuss in the form given here would lead to severe vulnerabilities of the schemes. 1 The reader is encouraged to start by reading the paper on Why textbook ElGamal and RSA encryption are insecure. 2 Our discussion falls into two components: in the first we exploit the difficulty of factoring integers into their prime factors. In the second we use large cyclic groups in order to establish secrets known only to the participating parties. 1 And if those sentences read like just another of those painful "Disclaimers" one finds everywhere, I apologise.

IEEE Access

Nowadays, enormous data is being generated by the interconnected digital world, and the need to secure data is highly demanding. Advancement in technologies and new innovative methods applied by the attackers play an instrumental role in breaching data security. Public key Cryptography provides a set of cryptographic algorithms in achieving data security through confidentiality, integrity and authentication. Among all cryptographic algorithms in general and public key cryptography in particular, RSA is one of the most widely used and applied algorithms. Since its inception, it is commonly being adopted in securing data across different domains such as cloud, image etc. Despite its importance and wide application, no such systematic and extensive survey exists in the literature. A systematic and thorough study of RSA based cryptography is presented in this work covering several domains. All the available works in this direction are divided into 11 different categories, viz, Hybrid, Parallel, Cloud, Image, Multiple-Keys, Chinese-Remainder-Theorem-based, Digital-Signatures, K-Nearest-Theorem-based, Batch, Wireless, and Core-Modifications. This study aims to explore RSA-based cryptosystems, either modifications in core RSA or application of enhanced RSA across different domains, systematically categorizing in various categories and eventually providing finding and indications. As a result, this study will guide researchers and practitioners in understanding the past and present status of RSA cryptography and the possibility of its applications in other domains.

Elliptic curve cryptography (ECC) is an approach to public key Cryptography based on the algebraic structure of Elliptic curves over finite field. Elliptic curves are also used in several integer factorization algorithms that have applications in cryptography. An elliptic curve in cryptography was suggested independently by Neal Koblitz and Victor S. Millar in 1985. Elliptic curve cryptography algorithms entered wide use in 2004 to 2005.Cryptography comes from Greek words meaning “hidden writing”. Cryptography converts readable data or clear text into encoded data called cipher text. By definition cryptography is the science of hiding information so that unauthorized users cannot read it. It involves Encryption and decryption of messages. Encryption is the process of converting a Plain text into cipher text and decryption is the process of getting back the original Message from the encrypted text. ECC is a newer approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields, and considered a s a marvelous technique with low key size for the use r, and hard exponential time challenge for an intruder to break into the system. In ECC a 16 0 bits key, provides the same security as RSA [1] 1024 bits key, thus lower computation f aster cryptographic power is required. The advantage of elliptic curve cryptosystems is the absence of sub exponential time algorithms, for attack. As ECC uses less key size to provide more security, and for this advantage it is used to perform operations, running on smaller chips or m ore compact software. The public key cryptography- based r emote authentication schemes are not suitable for mobile devices, because of the limitation in the bandwidth, computational strength, power availability or storage in mobile devices. Elliptic Curve cryptography is very difficult to understand by attacker because it relies on Elliptic Curve Discrete Logarithm Problem known as ECDLP. So it is difficult to break.

From September 25th till September 30th, 2011, the Dagstuhl Seminar 11391 about "Public-Key Cryptography" took place at Schloss Dagstuhl. The meeting hosted 33 international researchers and incited active discussions about recent developments in this area. Seminar 25.-30. September, 2011 -www.dagstuhl.de/11391 1998 ACM Subject Classification D.4.6 Security and Protection

Technical Report of National Institute of Science and Technology (NIST), India, Summer Research Program, 2013 , 2013

To overcome the problems faced in symmetric key algorithms, people have chosen Asymmetric Key algorithms for communication. Communication with Asymmetric algorithms will give us transmission of information without exchanging the key. Public-key cryptography refers to a cryptographic system requiring two separate keys, one of which is secret and one of which is public. Public-key cryptography is widely used. It is an approach used by many cryptographic algorithms and cryptosystems. It underpins such Internet standards as Transport Layer Security (TLS), PGP, and GPG. RSA and Diffie–Hellman key exchange are the most widely used public key distribution systems, while the Digital Signature Algorithm is the most widely used digital signature system. In this report we are mainly concentrating on some asymmetric algorithms which are mostly used. They are RSA cryptosystem and ElGamal Cryptosystem. It also gives brief mathematical explanations. The RSA algorithm is the most commonly used encryption and authentication algorithm and is included as part of the Web browsers from Microsoft and Netscape.RSA is an algorithm for public-key cryptography that is based on the presumed difficulty of factoring large integers, the factoring problem.. The RSA algorithm involves three steps: key generation, encryption and decryption. In this we mainly concentrate on algorithms for Primality Testing, Extended Euclidian’s algorithm, Modular Exponentiation solving algorithm, etc. ElGamal System is a public-key cryptosystem based on the discrete logarithm problem. It consists of both encryption and Signature algorithms. ElGamal encryption is used in the free GNU Privacy Guard software, recent versions of PGP, and other cryptosystems. ElGamal encryption consists of three components: the key generator, the encryption algorithm, and the decryption algorithm. In this we concentrate on the algorithms Cyclic Groups, Modular Exponentiation solving algorithms etc.