NTRU is a public key cryptosystem, which was first introduced in 1996. It is a ring-based cryptosystem and its security relies on the complexity of a well-known lattice problem, i.e. shortest vector problem (SVP). There is no efficient algorithm known to solve SVP exactly in arbitrary high dimensional lattices. However, approximate solutions to SVP can be found by lattice reduction algorithms. LLL is the first polynomial time algorithm that finds reasonable short vectors of a lattice. The best known attacks on the NTRU cryptosystem are lattice attacks. In these attacks, the lattice constructed by the public key of the system is used to find the private key. The target vector, which includes private key of the system is one of the short vectors of the NTRU lattice. In this thesis, we study NTRU cryptosystem and lattice attacks on NTRU. Also, we applied an attack to a small dimensional NTRU lattice.

In this paper, new software and hardware designs for the NTRU Public Key Cryptosystem are proposed. The first design attempts to improve NTRU's polynomial multiplication through applying techniques from the Chinese Remainder Theorem (CRT) to the convolution algorithm. Although the application of CRT shows promise for the creation of the inverse polynomials in the setup procedure, it does not provide any benefits to the procedures that are critical to the performance of NTRU (public key creation, encryption, and decryption). This research has identified that this is due to the small coefficients of one of the operands, which can be a common misunderstanding. The second design focuses on improving the performance of the polynomial multiplications within NTRU's key creation, encryption, and decryption procedures through hardware. This design exploits the inherent parallelism within a polynomial multiplication to make scalability possible. The advantage scalability provides is that it allows the user to customize the design for low and high power applications. In addition, the support for arbitrary precision allows the user to meet the desired security level. The third design utilizes the Montgomery Multiplication algorithm to develop an unified architecture that can perform a modular multiplication for GF(p) and GF(2^k) and a polynomial multiplication for NTRU. The unified design only requires an additional 10 gates in order for the Montgomery Multiplier core to compute the polynomial multiplication for NTRU. However, this added support for NTRU presents some restrictions on the supported lengths of the moduli and on the chosen value for the residue for the GF(p) and GF(2^k) cases. Despite these restrictions, this unified architecture is now capable of supporting public key operations for the majority of Public-Key Cryptosystems.

Národný inštitút pre štandardy a technológie (NIST) zahájil proces na získanie, vyhodnotenie a štandardizáciu jedného alebo viacerých kryptografických algoritmov využívajúcich verejný kľúč prostredníctvom verejnej súťaže. Cieľom tejto dimplomovej práce je naštudovať dostupné postkvantové algoritmy pre ustanovenie kľúča, ktoré boli zverejnené v treťom kole tejto súťaže. Po dôkladnej analýze a porovnaní bol jeden zo študovaných algoritmov implementovaný s využitím knižníc dostupných pre daný algoritmus, následne bol program optimalizovaný a zdokumentovaný.

This thesis focuses on issues related to authentication in low-cost radio frequency identification technology, more commonly referred to as RFID. This technology it is often referred to as the next technological revolution after the Internet. However, due to the very limited resources in terms of computation, memory and energy on RFID tags, conventional security algorithms cannot be implemented on low-cost RFID tags making security and privacy an important research subject today. First of all, we investigate the scalability in low-cost RFID systems by developing a ns-3 module to simulate the universal low-cost RFID standard EPC Class-1 Generation-2 in order to establish a strict framework for secure identification in low-cost RFID systems. We show that, the symmetrical key cryptography is excluded from being used in any scalable low-cost RFID standard. Then, we propose a scalable authentification protocol based on our adaptation of the famous public key cryptosystem NTRU. This protocol is specially designed for low-cost RFID systems, it can be efficiently implemented into low-cost tags. Finally, we consider the zero-knowledge identification i.e. when the no secret sharing between the tag and the reader is needed. Such identification approaches are very helpful in many RFID applications when the tag changes constantly the field of administration. We propose two lightweight zero-knowledge identification approaches based on GPS and randomized GPS schemes. The proposed approaches consist in storing in the back-end precomputed values in the form of coupons. So, the GPS-based variant can be private and the number of coupons can be much higher than in other approaches thus leading to higher resistance to denial of service attacks for cheaper tags