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26
Software Implementation of Elliptic Curve Cryptography Over Binary Fields
, 2000
"... This paper presents an extensive and careful study of the software implementation on workstations of the NISTrecommended elliptic curves over binary fields. We also present the results of our implementation in C on a Pentium II 400 MHz workstation. ..."
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Cited by 147 (9 self)
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This paper presents an extensive and careful study of the software implementation on workstations of the NISTrecommended elliptic curves over binary fields. We also present the results of our implementation in C on a Pentium II 400 MHz workstation.
A scalable and unified multiplier architecture for finite fields GF(p) and GF(2 m
 and GF (2 m ). In Cryptographic Hardware and Embedded Systems — CHES 2000, LNCS
, 2000
"... We describe a scalable and unified architecture for a Montgomery multiplication module which operates in both types of finite fields GF(p) and GF(2m). The unified architecture requires only slightly more area than that of the multiplier architecture for the field GF(p). The multiplier is scalable,wh ..."
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Cited by 44 (12 self)
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We describe a scalable and unified architecture for a Montgomery multiplication module which operates in both types of finite fields GF(p) and GF(2m). The unified architecture requires only slightly more area than that of the multiplier architecture for the field GF(p). The multiplier is scalable,which means that a fixedarea multiplication module can handle operands of any size,and also,the wordsize can be selected based on the area and performance requirements. We utilize the concurrency in the Montgomery multiplication operation by employing a pipelining design methodology. We also describe a scalable and unified adder module to carry out concomitant operations in our implementation of the Montgomery multiplication. The upper limit on the precision of the scalable and unified Montgomery multiplier is dictated only by the available memory to store the operands and internal results,and the module is capable of performing infiniteprecision Montgomery multiplication in both types of finite fields. Key Words: Prime fields,binary extension fields,multiplication,Montgomery multiplication, scalability,hardware implementation.
A Scalable Architecture for Montgomery Multiplication
 Lecture Notes in Computer Science
, 1999
"... This paper describes the methodology and design of a scalable Montgomery multiplication module. There is no limitation on the maximum number of bits manipulated by the multiplier, and the selection of the wordsize is made according to the available area and/or desired performance. We describe t ..."
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Cited by 41 (7 self)
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This paper describes the methodology and design of a scalable Montgomery multiplication module. There is no limitation on the maximum number of bits manipulated by the multiplier, and the selection of the wordsize is made according to the available area and/or desired performance. We describe the general view of the new architecture, analyze hardware organization for its parallel computation, and discuss design tradeo#s which are useful to identify the best hardware configuration.
Faster Algorithms for String Matching Problems: Matching the Convolution Bound
 In Proceedings of the 39th Symposium on Foundations of Computer Science
, 1998
"... In this paper we give a randomized O(n log n)time algorithm for the string matching with don't cares problem. This improves the FischerPaterson bound [10] from 1974 and answers the open problem posed (among others) by Weiner [30] and Galil [11]. Using the same technique, we give an O(n log n)t ..."
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Cited by 30 (5 self)
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In this paper we give a randomized O(n log n)time algorithm for the string matching with don't cares problem. This improves the FischerPaterson bound [10] from 1974 and answers the open problem posed (among others) by Weiner [30] and Galil [11]. Using the same technique, we give an O(n log n)time algorithm for other problems, including subset matching and tree pattern matching [15, 21, 9, 7, 17] and (general) approximate threshold matching [28, 17]. As this bound essentially matches the complexity of computing of the Fast Fourier Transform which is the only known technique for solving problems of this type, it is likely that the algorithms are in fact optimal. Additionally, the technique used for the threshold matching problem can be applied to the online version of this problem, in which we are allowed to preprocess the text and require to process the pattern in time sublinear in the text length. This result involves an interesting variant of the KarpRabin fingerprint m...
Instruction Set Extensions for Fast Arithmetic in Finite Fields GF(p) and GF(2m)
 CRYPTOGRAPHIC HARDWARE AND EMBEDDED SYSTEMS — CHES 2004
, 2004
"... Abstract. Instruction set extensions are a small number of custom instructions specifically designed to accelerate the processing of a given kind of workload such as multimedia or cryptography. Enhancing a generalpurpose RISC processor with a few applicationspecific instructions to facilitate the ..."
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Cited by 14 (6 self)
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Abstract. Instruction set extensions are a small number of custom instructions specifically designed to accelerate the processing of a given kind of workload such as multimedia or cryptography. Enhancing a generalpurpose RISC processor with a few applicationspecific instructions to facilitate the inner loop operations of publickey cryptosystems can result in a significant performance gain. In this paper we introduce a set of five custom instructions to accelerate arithmetic operations in finite fields GF(p) and GF(2^m). The custom instructions can be easily integrated into a standard RISC architecture like MIPS32 and require only little extra hardware. Our experimental results show that an extended MIPS32 core is able to perform an elliptic curve scalar multiplication over a 192bit prime field in 36 msec, assuming a clock speed of 33 MHz. An elliptic curve scalar multiplication over the binary field GF(2^191) takes only 21 msec, which is approximately six times faster than a software implementation on a standard MIPS32 processor.
Instruction Set Extension for Fast Elliptic Curve Cryptography Over Binary Finite Fields GF(2m)
 IN PROCEEDINGS OF THE 14TH IEEE INTERNATIONAL CONFERENCE ON APPLICATIONSPECIFIC SYSTEMS, ARCHITECTURES AND PROCESSORS (ASAP 2003)
, 2003
"... The performance of elliptic curve (EC) cryptosystems depends essentially on efficient arithmetic in the underlying finite field. Binary finite fields GF(2m) have the advantage of “carryfree” addition. Multiplication, on the other hand, is rather costly since polynomial arithmetic is not supported b ..."
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Cited by 13 (7 self)
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The performance of elliptic curve (EC) cryptosystems depends essentially on efficient arithmetic in the underlying finite field. Binary finite fields GF(2m) have the advantage of “carryfree” addition. Multiplication, on the other hand, is rather costly since polynomial arithmetic is not supported by generalpurpose processors. In this paper we propose a combined hardware/software approach to overcome this problem. First, we outline that multiplication of binary polynomials can be easily integrated into a multiplier datapath for integers without significant additional hardware. Then, we present new algorithms for multipleprecision arithmetic in GF(2m) based on the availability of an instruction for singleprecision multiplication of binary polynomials. The proposed hardware/software approach is considerably faster than a “conventional” software implementation and well suited for constrained devices like smart cards. Our experimental results show that an enhanced 16bit RISC processor is able to generate a 191bit ECDSA signature in less than 650 msec when the core is clocked at 5 MHz.
Parallel Montgomery Multiplication in GF(2 k ) Using Trinomial Residue Arithmetic
 In 17th IEEE Symposium on Computer Arithmetic (ARITH05
, 2005
"... We propose the first general multiplication algorithm in GF(2 k) with a subquadratic area complexity of O(k 8/5) = O(k 1.6). Using the Chinese Remainder Theorem, we represent the elements of GF(2 k); i.e. the polynomials in GF(2)[X] of degree at most k − 1, by their remainder modulo a set of n pair ..."
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Cited by 7 (0 self)
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We propose the first general multiplication algorithm in GF(2 k) with a subquadratic area complexity of O(k 8/5) = O(k 1.6). Using the Chinese Remainder Theorem, we represent the elements of GF(2 k); i.e. the polynomials in GF(2)[X] of degree at most k − 1, by their remainder modulo a set of n pairwise prime trinomials, T1,..., Tn, of degree d and such that nd ≥ k. Our algorithm is based on Montgomery’s multiplication applied to the ring formed by the direct product of the trinomials.
Software multiplication using Gaussian normal bases
 IEEE Trans. Comput
, 2006
"... Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over binary fields F2m. In this paper we present new software algorithms for efficient multiplication over F2m that use a Gaussian normal ..."
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Cited by 6 (2 self)
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Fast algorithms for multiplication in finite fields are required for several cryptographic applications, in particular for implementing elliptic curve operations over binary fields F2m. In this paper we present new software algorithms for efficient multiplication over F2m that use a Gaussian normal basis representation. Two approaches are presented, direct normal basis multiplication, and a method that exploits a mapping to a ring where fast polynomialbased techniques can be employed. Our analysis including experimental results on an Intel Pentium family processor shows that the new algorithms are faster and can use memory more efficiently than previous methods. Despite significant improvements, we conclude that the penalty in multiplication is still sufficiently large to discourage the use of normal bases in software implementations of elliptic curve systems. Key words Multiplication in F2 m, Gaussian normal basis, elliptic curve cryptography. 1
Highspeed Software Multiplication in F 2 m
 International Conference in Cryptology in India–INDOCRYPT 2000, volume 1977 of Lecture
, 2000
"... In this paper we describe an efficient algorithm for multiplication in F_2m , where the field elements of F_2m are represented in standard polynomial basis. The proposed algorithm can be used in practical software implementations of elliptic curve cryptography. Our timing results, on several platfor ..."
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Cited by 3 (1 self)
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In this paper we describe an efficient algorithm for multiplication in F_2m , where the field elements of F_2m are represented in standard polynomial basis. The proposed algorithm can be used in practical software implementations of elliptic curve cryptography. Our timing results, on several platforms, show that the new method is significantly faster than the "shiftandadd" method.
A Simple Architectural Enhancement for Fast and Flexible Elliptic Curve Cryptography Over Binary Finite Fields GF(2m)
 IN PROCEEDINGS OF THE NINTH ASIAPACIFIC CONFERENCE ON ADVANCES IN COMPUTER SYSTEMS ARCHITECTURE — ACSAC 2004
, 2004
"... Abstract. Mobile and wireless devices like cell phones and networkenhanced PDAs have become increasingly popular in recent years. The security of data transmitted via these devices is a topic of growing importance and methods of publickey cryptography are able to satisfy this need. Elliptic curve c ..."
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Cited by 3 (2 self)
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Abstract. Mobile and wireless devices like cell phones and networkenhanced PDAs have become increasingly popular in recent years. The security of data transmitted via these devices is a topic of growing importance and methods of publickey cryptography are able to satisfy this need. Elliptic curve cryptography (ECC) is especially attractive for devices which have restrictions in terms of computing power and energy supply. The efficiency of ECC implementations is highly dependent on the performance of arithmetic operations in the underlying finite field. This work presents a simple architectural enhancement to a generalpurpose processor core which facilitates arithmetic operations in binary finite fields GF(2 m). A custom instruction for a multiply step for binary polynomials has been integrated into a SPARC V8 core, which subsequently served to compare the merits of the enhancement for two different ECC implementations. One was tailored to the use of GF(2 191) with a fixed reduction polynomial. The tailored implementation was sped up by 90 % and its code size was reduced. The second implementation worked for arbitrary binary fields with a range of reduction polynomials. The flexible implementation was accelerated by a factor of nearly 10.