Modern Cryptography Volume 1

This open access book systematically explores the statistical characteristics of cryptographic systems, the computational complexity theory of cryptographic algorithms and the mathematical principles behind various encryption and decryption algorithms. The theory stems from technology. Based on Shannon's information theory, this book systematically introduces the information theory, statistical characteristics and computational complexity theory of public key cryptography, focusing on the three main algorithms of public key cryptography, RSA, discrete logarithm and elliptic curve cryptosystem. It aims to indicate what it is and why it is. It systematically simplifies and combs the theory and technology of lattice cryptography, which is the greatest feature of this book. It requires a good knowledge in algebra, number theory and probability statistics for readers to read this book. The senior students majoring in mathematics, compulsory for cryptography and science and engineering postgraduates will find this book helpful. It can also be used as the main reference book for researchers in cryptography and cryptographic engineering areas

Encryption, Elliptic Curves, and the Symmetries of Differential Equations

In cryptography, encryption is the process of encoding messages in such a way that only authorized parties can access them. The intended information, referred to as plaintext, is encrypted using an encryption algorithm, generating ciphertext that can only be read if decrypted. Public key cryptography, or asymmetric cryptography, is any cryptographic system that uses pairs of keys: public keys which may be disseminated widely, and private keys which are known only to the owner. In a public key encryption system, any person can encrypt a message using the public key, but such a message can be decrypted only with the private key. Elliptic curve cryptography (ECC) is a particularly powerful approach to public-key cryptography based on tori or more precisely elliptic curves. The purpose of this talk is to discuss the mathematics employed in elliptic curve encryption which is based on the algebraic structure of elliptic curves, in particular on the ability to add points. Such group structure on a torus is evident if we represent it as a fundamental domain in the complex plane with its edges identified. Once the group structure has been defined in the complex plane, the group structure on a torus is evident. In turn, an elliptic curve is parameterized over the complex plane by the Weierstrass elliptic function. Moreover, the Weierstrass elliptic function allows to identify the defining quantities of a torus with those of an elliptic curve using modular forms

Modern Cryptography Volume 1

This open access book systematically explores the statistical characteristics of cryptographic systems, the computational complexity theory of cryptographic algorithms and the mathematical principles behind various encryption and decryption algorithms. The theory stems from technology. Based on Shannon's information theory, this book systematically introduces the information theory, statistical characteristics and computational complexity theory of public key cryptography, focusing on the three main algorithms of public key cryptography, RSA, discrete logarithm and elliptic curve cryptosystem. It aims to indicate what it is and why it is. It systematically simplifies and combs the theory and technology of lattice cryptography, which is the greatest feature of this book. It requires a good knowledge in algebra, number theory and probability statistics for readers to read this book. The senior students majoring in mathematics, compulsory for cryptography and science and engineering postgraduates will find this book helpful. It can also be used as the main reference book for researchers in cryptography and cryptographic engineering areas

Public key cryptography based on tropical algebra

We analyse some public keys cryptography in the classical algebra and tropical algebra. Currently one of the most secure system that is used is public key cryptography, which is based on discrete logarithm problem. The Dilfie-Helman public key and Stickel’s key ex-change protocol are the examples of the application of discrete logarithm problem in public key cryptography. This thesis will examine the possibilities of public key cryptography implemented within tropical mathematics. A tropical version of Stickel’s key exchange protocol was suggested by Grigoriev and Sphilrain We suggest some modifications of this scheme use commuting matrices in tropical algebra and discuss some possibilities of at- tacks on them. We also generalise Kotov and Ushakov’s attack and implement in our new protocols. In 2019, Grigoriev and Sphilrain [14] generated two new public key exchange protocols based on semidirect product. In this thesis we use some properties of CSR and ultimate periodicity in tropical algebra to construct an efficient attack on one of the protocols suggested in that pape

Data Security Using Stegnography and Quantum Cryptography

Stegnography is the technique of hiding confidential information within any media. In recent years variousstegnography methods have been proposed to make data more secure. At the same time differentsteganalysis methods have also evolved. The number of attacks used by the steganalyst has only multipliedover the years. Various tools for detecting hidden informations are easily available over the internet, sosecuring data from steganalyst is still considered a major challenge. While various work have been done toimprove the existing algorithms and also new algorithms have been proposed to make data behind theimage more secure. We have still been using the same public key cryptography like Deffie-Hellman andRSA for key negotiation which is vulnerable to both technological progress of computing power andevolution in mathematics, so in this paper we have proposed use of quantum cryptography along withstegnography. The use of this combination will create key distribution schemes that are uninterceptable thusproviding our data a perfect security.Keywords: Stegnography, Steganalysis, Steganalyst, Quantum Cryptography

Human and Technical Factors in the Adoption of Quantum Cryptographic Algorithms

The purpose of this research is to understand what factors would cause users to choose quantum key distribution (QKD) over other methods of cryptography. An Advanced Encryption Standard (AES) key can be exchanged through communication using the Rivest, Shamir, Adleman (RSA) cryptographic algorithm, QKD, or post-quantum cryptography (PQC). QKD relies on quantum physics where RSA and PQC use complex mathematics to encrypt data. The BB84 quantum cryptographic protocol involves communication over a quantum channel and a public channel. The quantum channel can be technically attacked by beamsplitting or intercept/resend. QKD, like other forms of cryptography, is vulnerable to social attacks such as industrial espionage. QKD products can transmit over maximum distances ranging from 40 km up to 150 km with key rates as low as 1.4 kb/s up to at least 300 kb/s. A survey and focus group discussion with a defense contracting company revealed that while nobody fully trusts current security systems, they are more concerned about social engineering attacks before attacks on cryptography. The company is not interested in implementing QKD unless the range capabilities are improved or there is regulation requiring them to use it

Cryptography in the digital age

Despite it not being an armed conflict between nations, there is a war that has been waged for over 5000 years and is still being fought today. Battles have been won and lost by both sides. The battlefield is the world of cryptography. Our combatants are cryptographers, personnel who make secret codes; and cryptanalysts, personnel who try to break the secret codes; In this thesis, we examine public-key or asymmetric cryptography, the are of writing or deciphering secret codes or ciphers. We begin by taking a brief look at the overall history of cryptography. Our primary focus involves studying the mathematics behind today\u27s public-key cryptographic methods such as the theory of congruences by Carl Friedrich Gauss, Fermat\u27s little theorem, Euler\u27s phi-function, primitive roots and indices, and elliptic curves over finite fields. Once we have explore the preliminaries we will consider some of the more popular methods of encryption and decryption, for example RSA. We not only discuss how to encrypt and decrypt plain text using these methods but explain why it is hard to break the encrypted text. We conclude our study by inspecting the shortfall of these ciphers, techniques used to break the encryption, and what the future possibly holds

Cryptography: Mathematical Advancements on Cyber Security

The origin of cryptography, the study of encoding and decoding messages, dates back to ancient times around 1900 BC. The ancient Egyptians enlisted the use of basic encryption techniques to conceal personal information. Eventually, the realm of cryptography grew to include the concealment of more important information, and cryptography quickly became the backbone of cyber security. Many companies today use encryption to protect online data, and the government even uses encryption to conceal confidential information. Mathematics played a huge role in advancing the methods of cryptography. By looking at the math behind the most basic methods to the newest methods of cryptography, one can learn how cryptography has advanced and will continue to advance