There are several reasons for presenting lattices in this book. First, there are hard computational problems on lattices that have been used as a building block for public key cryptosystems (e.g., the Goldreich-Goldwasser-Halevi (GGH) cryptosystem, the NTRU cryptosystem, the Ajtai-Dwork cryptosystem, and the LWE cryptosystem); however, we do not present these applications in this book. Second, ...

This thesis analyzes the security and e ciency of public key cryptosystems. New attacks for several cryptosystems are proposed and the e ectiveness of the attacks is evaluated. Furthermore, solutions are given to several unsolved problems in computational number theory and algebraic geometry theory that are closely related to the security of public key cryptosystems. Moreover, new calculation m...

Public-key cryptosystems are one of the basic building blocks of modern cryptographic systems. Therefore, it is important that they allow to be implemented resistant to several types of side-channel attacks. In this survey paper, we give an overview of the possible vulnerabilities of implementations of public-key cryptosystems that have been submitted to the NESSIE project and we additionally d...

This paper presents three methods for strengthening public key cryptosys-tems in such a way that they become secure against adaptively chosen ciphertext attacks. In an adaptively chosen ciphertext attack, an attacker can query the deciphering algorithm with any ciphertexts, except for the exact object cipher-text to be cryptanalyzed. The rst strengthening method is based on the use of one-way h...

Nonabelian group-based public key cryptography is a relatively new and exciting research field. Rapidly increasing computing power and the futurity quantum computers [52] that have since led to, the security of public key cryptosystems in use today, will be questioned. Research in new cryptographic methods is also imperative. Research on nonabelian group-based cryptosystems will become one of c...

Since the introduction of public key cryptography in the 70’s [1], many cryptosystems have been proposed and many cryptographic schemes have been broken. The most used cryptosystems rely on number theory problem like the factorization problem [3] and the discrete logarithm over suitable group [2]. The McEliece cryptosystem [5] and the Neiderreiter variante [6] rely on coding theory, they are on...

The basic Merkle-Hellman additive trapdoor knapsack public-key cryptosystem was recently shown to be insecure, and attacks have also been developed on stronger variants of it, such as the Graham-Shamir system and the iterated knapsack cryptosystem. This paper shows that some simple variants of another Merkle-Hellman system, the multiplicative knapsack cryptosystem, are insecure. It is also show...

Digital signature schemes are crucial for applications in electronic commerce. The effectiveness and security of a digital signature scheme rely on its underlying public key cryptosystem. Trapdoor functions are central to public key cryptosystems. However, the modular exponentiation for RSA or the discrete logarithms for ElGamal/DSA/ECC, as the choice of the trapdoor functions, are relatively s...

Most practical public-key cryptosystems are based on modular exponentiation. A modular exponentiation is composed of repeated modular multiplications. Several methods have been proposed to reduce the execution time of a modular exponentiation, essentially aiming to reduce the execution time of each modular multiplication. The goal of this paper is to investigate three different integer multipli...