This thesis examines the Mordell-Weil group for application in cryptography. This approach has recently been proposed by Gerhard Frey. The use of the Mordell-Weil group for discrete logarithm schemes is a variant of elliptic curve cryptosystems. We extended the original idea by Frey with the goal of a performance improvement. The arithmetic complexity using the Mordell-Weil group will be compared to ordinary elliptic curve cryptosystems. The main goals of this thesis are (1) to investigate the algorithmic complexity of Mordell-Weil cryptosystems relative to elliptic curve cryptosystems; (2) the appropriate selection of the group parameters for a successful adaptation to different platforms; (3) a C++ library which makes it possible to easily use this algebra for cryptographic systems based on groups; and (4) to obtain software performance measures for the new cryptosystem. Point multiplication, the crucial operation for elliptic curve cryptosystems, is more than 20% less complex in the Mordell-Weil group than in an ordinary elliptic curve while preserving the same level of security. We show how to further improve the system such that it is particularly suited to 32-bit and 16-bit hardware platforms. The speed-up of the Mordell-Weil group approach comes at the cost of a slightly larger bit-size that is needed to represent a curve point and a more costly curve generation.

For over two decades, mathematicians and cryptologists have evaluated and presented the theoretical performance of Elliptic-curve scalar point-multiplication in projective geometry. Because computation in projective domain is composed of a wide array of formulations and computing optimizations, there is not a comprehensive performance comparison of point-multiplication using projective transformation available to verify its realistic efficiency in 64-bit x86 computing platforms. Today, research on explicit mathematical formulations in projective domain continues to excel by seeking higher computational efficiency and ease of realization. An explicit performance evaluation will help implementers choose better implementation methods and improve Elliptic-curve scalar point-multiplication. This paper was founded on the practical solution that obtaining realistic performance figures should be based on more precise computational cost metrics and specific computing platforms. As part of that solution, an empirical performance benchmark comparison between two approaches implementing projective Elliptic-curve scalar point-multiplication will be presented to provide the selection of, and subsequently ways to improve scalar point-multiplication technology executing in a 64-bit x86 runtime environment.

Elliptic curve cryptography (ECC) is becoming the algorithm of choice for digital signature generation and authentication in embedded context. However, performance of ECC and the underlying modular arithmetic on embedded processors remains a concern. At the same time, more complex system-on-chip platforms with multiple heterogeneous cores are commonly available in mobile phones and other embedded devices. In this work we investigate the design space for ECC on TI's OMAP 3530 platform, with a focus of utilizing the on-chip DSP core to improve the performance and efficiency of ECC point multiplication on the target platform. We examine multiple aspects of ECC and heterogeneous design such as algorithm-level choices for elliptic curve operations and the effect of interprocessor communication overhead on the design partitioning. We observe how the limitations of the platform constrict the design space of ECC. However, by closely studying the platform and efficiently partitioning the design between the general purpose ARM core and the DSP, we demonstrate a significant speed-up of the resulting ECC implementation. Our system focused approach allows us to accurately measure the performance and power profiles of the resulting implementation. We conclude that heterogeneous multiprocessor design can significantly improve the performance and power consumption of ECC operations, but that the integration cost and the overhead of interprocessor communication cannot be ignored in any actual system. / Master of Science

Elliptic curve cryptography (ECC) has been introduced as a public-key cryptosystem, which offers smaller key sizes than the other known public-key systems at equivalent security level. The key size advantage of ECC provides faster computations, less memory consumption, less processing power and efficient bandwidth usage. These properties make ECC attractive especially for the next generation public-key cryptosystems. The implementation of ECC involves so many arithmetic operations / one of them is the elliptic curve point multiplication operation, which has a great influence on the performance of ECC protocols. In this thesis work, we have studied on elliptic curve point multiplication methods which are proposed by many researchers. The software implementations of these methods are developed in C programming language on Pentium 4 at 3 GHz. We have used NIST-recommended elliptic curves over prime and binary fields, by using efficient finite field arithmetic. We have then applied our elliptic curve point multiplication implementations to Elliptic Curve Digital Signature Algorithm (ECDSA), and compared different methods. The timing results are presented and comparisons with recent studies have been done.