Due to the intractability of the Discrete Logarithm Problem (DLP), it has been widely used in the field of cryptography and the security of several cryptosystems is based on the hardness of computation of DLP. In this paper, we start with the topics on Number Theory and Abstract Algebra as it will enable one to study the nature of discrete logarithms in a comprehensive way, and then, we concentrate on the application and computation of discrete logarithms. Application of discrete logarithms such as Diffie Hellman key exchange, ElGamal signature scheme, and several attacks over the DLP such as Baby-step Giant-step method, Silver Pohlig Hellman algorithm, etc have been analyzed. We also focus on the elliptic curve along with the discrete logarithm over the elliptic curve. Attacks for the elliptic curve discrete logarithm problem, ECDLP have been discussed. Moreover, the extension of several discrete logarithms-based protocols over the elliptic curve such as the elliptic curve digital signature algorithm, ECDSA have been discussed also.
Probabilistic Micropayments
(2022)
Probabilistic micropayments are important cryptography research topics in electronic commerce. The Probabilistic micropayments have the potential to be researched in order to obtain efficient algorithms with low transaction costs and high speeding computer power. To delve into the topic, it is vital to scrutinize the cryptographic preliminaries such as hash functions and digital signatures. This thesis investigates the important probabilistic methods based on a centralized or decentralized network. Firstly, centralized networks such as lottery-based tickets, Payword, coin-flipping, and MR2 are described, and an approach based on blind signatures is also discussed. Then, decentralized network methods such as MICROPAY3, a transferable scheme on the blockchain network, along with an efficient model for cryptocurrencies, are explained. Then we compare the different probabilistic micropayment methods by improving their drawback with a new technique. To set the results from the theoretical analysis of different methods into some context, we analyze the attacks that reduce the security and, therefore, the system’s efficiency. Particularly, we discuss various methods for detecting double-spending and eclipse attacks occurrence
In dieser Masterarbeit werden sichere steganografische sowie kryptografische Methoden vorgestellt, erläutert, untersucht und innerhalb eines eigens entwickelten Software-Prototypen mit intuitiver Benutzerschnittstelle kombiniert. Noch immer werden Menschenrechtsverteidigende in totalitären Systemen und anderen Krisengebieten systematisch verfolgt, inhaftiert, gefoltert oder sogar exekutiert, weil ihre digital gespeicherten Daten eine antitotalitäre und investigative Tätigkeit beweisen. Die in dieser Arbeit gesammelten Erkenntnisse sowie der darauf basierende Prototyp sollen zu einem besseren Schutz dieser Menschen beitragen.
Fermat proposed fermat’s little theorem in 1640, but a proof was not officially published until 1736. In this thesis paper, we mainly focus on different proofs of fermat’s little theorem like combinatorial proof by counting necklaces, multinomial proofs, proof by modular arithmetic, dynamical systems proof, group theory proof etc. We also concentrate on the generalizations of fermat’s little theorem given by Euler and Laplace. Euler was the first scientist to prove the fermat’s little theorem. We will also go through three different proofs given by Euler for fermat’s little theorem. This theorem has many applications in the field of mathematics and cryptography. We focus on applications of fermat’s little theorem in cryptography like primality testing and publickey cryptography. Primality test is used to determine if the given number n is a prime number or composite number. In this paper, we also concentrate on fermat primality test and Miller-Rabin primality test, which is an extension of fermat primality test. We also discuss the most widely used public-key cryptosystem i.e, the RSA Algorithm, named after its developers R. Rivest, A. Shamir, and L. Adleman. The algorithm was invented in 1978 and depends heavily on fermat’s little theorem.
The emerging Internet of Things (IoT) technology interconnects billions of embedded devices with each other. These embedded devices are internet-enabled, which collect, share, and analyze data without any human interventions. The integration of IoT technology into the human environment, such as industries, agriculture, and health sectors, is expected to improve the way of life and businesses. The emerging technology possesses challenges and numerous
security threats. On these grounds, it is a must to strengthen the security of IoT technology to avoid any compromise, which affects human life. In contrast to implementing traditional cryptosystems on IoT devices, an elliptic curve cryptosystem (ECC) is used to meet the limited resources of the devices. ECC is an elliptic curve-based public-key cryptography which provides equivalent security with shorter key size compared to other cryptosystems such as Rivest–Shamir–Adleman (RSA). The security of an ECC hinges on the hardness to solve the elliptic curve discrete logarithm problem (ECDLP). ECC is faster and easier to implement and also consumes less power and bandwidth. ECC is incorporated in internationally recognized standards for lightweight applications due to the
benefits ECC provides.
In der vorliegenden Masterarbeit werden die neuesten Entwicklungen kryptoanalytischer Verfahren zur Lösung des diskreten Logarithmus in der Elliptischen-Kurven-Kryptographie zusammengefasst, um Schlussfolgerungen über die Sicherheit aktuell verwendbarer Schlüssellängen aufzustellen. Dabei werden auch derzeit eingesetzte Hardwarelösungen betrachtet, die ebenfalls die Sicherheit aktuell verwendbarer Schlüssellängen beeinflussen