Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other public-key schemes based on the discrete logarithm in finite fields and the integer factorization problem and are thus attractive for many applications. This thesis describes an implementation of a crypto engine based on elliptic curves. The underlying algebraic structures are composite Galois fields GF((2 n) m) in a standard base representation. As a major new feature, the system is developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients, field order, or field representation. The thesis deals with the design and implementation of elliptic curve point multiplicationarchitectures. The architectures are described in VHDL and mapped to Xilinx FPGA devices. Architectures over Galois fields of different order and representation were implemented and compared. Area and timing measurements are provided for all architectures. It is shown that a full point multiplication on elliptic curves of real-world size can be implemented on commercially available FPGAs.

This contribution introduces a new class of multipliers for finite fields GF ((2 n ) 4 ). The architecture is based on a modified version of the Karatsuba-Ofman algorithm (KOA). By determining optimized field polynomials of degree four, the last stage of the KOA and the modulo reduction can be combined. This saves computation and area in VLSI implementations. The new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with KOA-based architectures with separate modulo reduction. The new multipliers lead to highly modular architectures an are thus well suited for VLSI implementations. Three types of field polynomials are introduced and conditions for their existence are established. For the small fields where n = 2; 3; : : : ; 8, which are of primary technical interest, optimized field polynomials were determined by an exhaustive search. For each field order, exact space and ti...

Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...

Multiplication with constant elements from Galois fields of characteristic two is the major arithmetic operation in Reed-Solomon encoders. This contribution describes two optimization algorithms which yield low complexity constant multipliers for Galois fields GF (2 n ). This seems to be the first reported systematic approach to this problem. We provide two locally optimum algorithms. Both algorithm are applied to fields GF (2 n ), n = 4; 5; : : : ; 16. The performance of the algorithms is compared to a straightforward approach. It is found that the optimization yields considerable improvements in the computational complexity. For the important field GF (2 8 ), the average number of modulo 2 additions (or XOR gates) is reduced by 40% compared to a straightforward implementation. 1 Introduction Reed-Solomon error correction codes are widely used in todays communication systems. Numerous application such as optical storage systems, digital TV, and space communication systems use R...

this paper, a low-complexity Programmable Cellular Automata (PCA) based versatile modular multiplier in GF(2 ) is presented. The proposed versatile multiplier increases flexibility in using the same multiplier in different security environments, and it reduces the user's cost. Moreover, the multiplier can be easily extended to high order of m for more security, and low-cost serial implementation is feasible in restricted computing environments, such as smart cards and wireless devices

University ofWaterloo

This contribution describes a comprehensive comparison of bit parallel finite field multipliers. Four different multipliers in standard, dual, and normal base together with a relatively new approach which uses composite fields are compared. Four different field orders from 2 8 to 2 32 are investigated. A high practical relevance is assured by using a highly automated design process and sea-of-gates chip in 0.8¯m technology as target hardware. The synthesis tool Synopsys is used for mapping and optimization. Unlike previous studies, quantitative results with respect to area and time performance are achieved. It is found that the new architecture requires the smallest number of gate equivalences. Dual and standard base multipliers require 30--40% more gates but have a somewhat smaller delay. The normal base multiplier has by far the highest gate consumption. It is concluded that the theoretical gate count is a valid estimate for the area requirement. 1 Introduction Many modern commu...

This thesis describes various efficient architectures for computation in Galois fields of the type GF(2^k).

The theory of elliptic curves is a classical topic in many branches of algebra and number theory, but recently it is receiving more attention in cryptography. An elliptic curve is a two-dimensional (planar) curve defined by an equation involving a cubic power of coordinate x and a square power of coordinate y. One class of these curves is