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37
Guide to Elliptic Curve Cryptography
, 2004
"... Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves ..."
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Cited by 369 (17 self)
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Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in publickey cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demonstrate why these systems provide relatively small block sizes, highspeed software and hardware implementations, and offer the highest strengthperkeybit of any known publickey scheme.
Fast Key Exchange with Elliptic Curve Systems
, 1995
"... The DiffieHellman key exchange algorithm can be implemented using the group of points on an elliptic curve over the field F 2 n . A software version of this using n = 155 can be optimized to achieve computation rates that are significantly faster than nonelliptic curve versions with a similar leve ..."
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Cited by 99 (2 self)
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The DiffieHellman key exchange algorithm can be implemented using the group of points on an elliptic curve over the field F 2 n . A software version of this using n = 155 can be optimized to achieve computation rates that are significantly faster than nonelliptic curve versions with a similar level of security. The fast computation of reciprocals in F 2 n is the key to the highly efficient implementation described here. March 31, 1995 Department of Computer Science The University of Arizona Tucson, AZ 1 Introduction The DiffieHellman key exchange algorithm [10] is a very useful method for initiating a conversation between two previously unintroduced parties. It relies on exponentiation in a large group, and the software implementation of the group operation is usually computationally intensive. The algorithm has been proposed as an Internet standard [13], and the benefit of an efficient implementation would be that it could be widely deployed across a variety of platforms, greatl...
Parameterized Complexity: A Framework for Systematically Confronting Computational Intractability
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1997
"... In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by ..."
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Cited by 72 (15 self)
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In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that kStep Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph kColoring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.
Parameterized Computational Feasibility
 Feasible Mathematics II
, 1994
"... Many natural computational problems have input consisting of two or more parts. For example, the input might consist of a graph and a positive integer. For many natural problems we may view one of the inputs as a parameter and study how the complexity of the problem varies if the parameter is he ..."
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Cited by 61 (20 self)
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Many natural computational problems have input consisting of two or more parts. For example, the input might consist of a graph and a positive integer. For many natural problems we may view one of the inputs as a parameter and study how the complexity of the problem varies if the parameter is held fixed. For many applications of computational problems involving such a parameter, only a small range of parameter values is of practical significance, so that fixedparameter complexity is a natural concern. In studying the complexity of such problems, it is therefore important to have a framework in which we can make qualitative distinctions about the contribution of the parameter to the complexity of the problem. In this paper we survey one such framework for investigating parameterized computational complexity and present a number of new results for this theory.
Finite Field Multiplier Using Redundant Representation
 IEEE Transactions on Computers
, 2002
"... This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our ar ..."
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Cited by 21 (1 self)
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This article presents simple and highly regular architectures for finite field multipliers using a redundant representation. The basic idea is to embed a finite field into a cyclotomic ring which has a basis with the elegant multiplicative structure of a cyclic group. One important feature of our architectures is that they provide areatime tradeoffs which enable us to implement the multipliers in a partialparallel/hybrid fashion. This hybrid architecture has great significance in its VLSI implementation in very large fields. The squaring operation using the redundant representation is simply a permutation of the coordinates. It is shown that when there is an optimal normal basis, the proposed bitserial and hybrid multiplier architectures have very low space complexity. Constant multiplication is also considered and is shown to have advantage in using the redundant representation. Index terms: Finite field arithmetic, cyclotomic ring, redundant set, normal basis, multiplier, squaring.
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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Cited by 19 (0 self)
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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.
A microcoded elliptic curve processor using FPGA technology
 IEEE Transactions on VLSI Systems
, 2002
"... Abstract—The implementation of a microcoded elliptic curve processor using fieldprogrammable gate array technology is described. This processor implements optimal normal basis field operations in P. The design is synthesized by a parameterized module generator, which can accommodate arbitrary and a ..."
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Cited by 17 (0 self)
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Abstract—The implementation of a microcoded elliptic curve processor using fieldprogrammable gate array technology is described. This processor implements optimal normal basis field operations in P. The design is synthesized by a parameterized module generator, which can accommodate arbitrary and also produce field multipliers with different speed/area tradeoffs. The control part of the processor is microcoded, enabling curve operations to be incorporated into the processor and hence reducing the chip’s I/O requirements. The microcoded approach also facilitates rapid development and algorithmic optimization: for example, projective and affine coordinates were supported using different microcode. The design was successfully tested on a Xilinx Virtex XCV10006 device and could perform an elliptic curve multiplication over the field P using affine and projective coordinates for aIIQISS and IUQ. Index Terms—Arithmetic, cryptography, Galois fields, microprogramming, public key cryptography, reconfigurable architectures. I.
FixedParameter Complexity and Cryptography
, 1993
"... . We discuss the issue of the parameterized computational complexity of a number of problems of interest in cryptography. We show that the problem of determining whether an ndigit number has a prime divisor less than or equal to n k can be solved in expected time f(k)n 3 by a randomized algo ..."
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Cited by 14 (11 self)
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. We discuss the issue of the parameterized computational complexity of a number of problems of interest in cryptography. We show that the problem of determining whether an ndigit number has a prime divisor less than or equal to n k can be solved in expected time f(k)n 3 by a randomized algorithm that employs elliptic curve factorization techniques (this result depends on an unproved but plausible numbertheoretic conjecture). An analogous computational problem concerning discrete logarithms is directly relevant to some proposed cryptosystem implementations. Our result suggests caution about implementations which fix a parameter such as the size or Hamming weight of keys. We show that several parameterized problems of relevance to cryptography, including kSubset Sum, kPerfect Code, and kSubset Product are likely to be intractable with respect to fixedparameter complexity. In particular, we show that they cannot be solved in time f(k)n ff , where ff is independent...
On Orders of Optimal Normal Basis Generators
 Math. Comp
, 1995
"... In this paper we give some computational results on the multiplicative orders of optimal normal basis generators in F2 n over F2 for n # 1200 whenever the complete factorization of 2  1 is known. Our results show that a subclass of optimal normal basis generators always have very high multiplic ..."
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Cited by 14 (6 self)
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In this paper we give some computational results on the multiplicative orders of optimal normal basis generators in F2 n over F2 for n # 1200 whenever the complete factorization of 2  1 is known. Our results show that a subclass of optimal normal basis generators always have very high multiplicative orders and are very often primitive. For a given optimal normal basis generator # in F2 n and an arbitrary integer e, we show that # can be computed in O(n v(e)) bit operations, where v(e) is the number of 1's in the binary representation of e.