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Elliptic curve cryptosystems on reconfigurable hardware
 MASTER’S THESIS, WORCESTER POLYTECHNIC INST
, 1998
"... 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 ..."
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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 publickey 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 realworld size can be implemented on commercially available FPGAs.
Efficient Multiplier Architectures for Galois Fields GF(2 4n )
 IEEE Transactions on Computers
, 1998
"... 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 KaratsubaOfman algorithm (KOA). By determining optimized field polynomials of degree four, the last stage of the KOA and the modulo reduction can b ..."
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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 KaratsubaOfman 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 KOAbased 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...
Normal Bases over Finite Fields
, 1993
"... 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 repr ..."
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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...
Optimized Arithmetic for ReedSolomon Encoders
 in 1997 IEEE International Symposium on Information Theory
, 1997
"... Multiplication with constant elements from Galois fields of characteristic two is the major arithmetic operation in ReedSolomon 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 ..."
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Cited by 7 (2 self)
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Multiplication with constant elements from Galois fields of characteristic two is the major arithmetic operation in ReedSolomon 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 ReedSolomon 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...
High Speed Architecture for Galois/Counter Mode of Operation (GCM)
 IACR PREPRINT ARCHIVE
, 2005
"... In this paper we present a fully pipelined high speed hardware architecture for Galois/Counter Mode of Operation (GCM) by analyzing the data dependencies in the GCM algorithm at the architecture level. We show that GCM encryption circuit and GCM authentication circuit have similar critical path dela ..."
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In this paper we present a fully pipelined high speed hardware architecture for Galois/Counter Mode of Operation (GCM) by analyzing the data dependencies in the GCM algorithm at the architecture level. We show that GCM encryption circuit and GCM authentication circuit have similar critical path delays resulting in an efficient pipeline structure. The proposed GCM architecture yields a throughput of 34 Gbps running at 271 MHz using a 0.18 µm CMOS standard cell library.
Efficient Cellular Automata Based Versatile Multiplier for GF(2^m)
 Journal of Information Science and Engineering
, 2002
"... this paper, a lowcomplexity 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 mu ..."
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this paper, a lowcomplexity 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 lowcost serial implementation is feasible in restricted computing environments, such as smart cards and wireless devices
Optimal Tower Fields,” in
 IEEE Transactions on Computers
"... Abstract — We introduce a new tower field representation, optimal tower fields (OTFs), that facilitates efficient finite field operations. The recursive direct inversion method we present has significantly lower complexity than the known best method for inversion in optimal extension fields (OEFs), ..."
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Abstract — We introduce a new tower field representation, optimal tower fields (OTFs), that facilitates efficient finite field operations. The recursive direct inversion method we present has significantly lower complexity than the known best method for inversion in optimal extension fields (OEFs), i.e., ItohTsujii’s inversion technique. The complexity of our inversion algorithm is shown to be O(m 2), significantly better than that of the ItohTsujii algorithm, i.e. O(m 2 (log 2 m)). This complexity is further improved to O(m log 2 3) by utilizing the KaratsubaOfman algorithm. In addition, we show that OTFs may be converted to OEF representation via a simple permutation of the coefficients, and hence OTF operations may be utilized to achieve the OEF arithmetic operations whenever a corresponding OTF representation exists. While the original OTF multiplication and squaring operations require slightly more additions than their OEF counterparts, due to the free conversion, both OTF operations may be achieved with the complexity of OEF operations. Index Terms — Optimal tower fields, OEF, finite fields, multiplication, inversion.
Finite field Multiplier Architectures for Cryptographic Applications
, 2000
"... University ofWaterloo ..."
A Comparative VLSI Synthesis of Finite Field Multipliers
 In 3rd International Symposium on Communication Theory and its Applications, Lake District
, 1995
"... 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 invest ..."
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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 seaofgates 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 3040% 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...
i Preface
"... This thesis describes various efficient architectures for computation in Galois fields of the type GF(2^k). ..."
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This thesis describes various efficient architectures for computation in Galois fields of the type GF(2^k).