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24
The Elliptic Curve Digital Signature Algorithm (ECDSA)
, 1999
"... The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA). It was accepted in 1999 as an ANSI standard, and was accepted in 2000 as IEEE and NIST standards. It was also accepted in 1998 as an ISO standard, and is under consideratio ..."
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Cited by 102 (5 self)
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The Elliptic Curve Digital Signature Algorithm (ECDSA) is the elliptic curve analogue of the Digital Signature Algorithm (DSA). It was accepted in 1999 as an ANSI standard, and was accepted in 2000 as IEEE and NIST standards. It was also accepted in 1998 as an ISO standard, and is under consideration for inclusion in some other ISO standards. Unlike the ordinary discrete logarithm problem and the integer factorization problem, no subexponentialtime algorithm is known for the elliptic curve discrete logarithm problem. For this reason, the strengthperkeybit is substantially greater in an algorithm that uses elliptic curves. This paper describes the ANSI X9.62 ECDSA, and discusses related security, implementation, and interoperability issues. Keywords: Signature schemes, elliptic curve cryptography, DSA, ECDSA.
Efficient arithmetic on Koblitz curves
 Designs, Codes, and Cryptography
, 2000
"... Abstract. It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the ..."
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Cited by 79 (0 self)
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Abstract. It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the elliptic scalar multiplication operation. Koblitz introduced a family of curves which admit especially fast elliptic scalar multiplication. His algorithm was later modified by Meier and Staffelbach. We give an improved version of the algorithm which runs 50 % faster than any previous version. It is based on a new kind of representation of an integer, analogous to certain kinds of binary expansions. We also outline further speedups using precomputation and storage.
An Overview of Elliptic Curve Cryptography
, 2000
"... Elliptic curve cryptography (ECC) was introduced by Victor Miller and Neal Koblitz in 1985. ECC proposed as an alternative to established publickey systems such as DSA and RSA, have recently gained a lot attention in industry and academia. The main reason for the attractiveness of ECC is the fact t ..."
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Cited by 29 (2 self)
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Elliptic curve cryptography (ECC) was introduced by Victor Miller and Neal Koblitz in 1985. ECC proposed as an alternative to established publickey systems such as DSA and RSA, have recently gained a lot attention in industry and academia. The main reason for the attractiveness of ECC is the fact that there is no subexponential algorithm known to solve the discrete logarithm problem on a properly chosen elliptic curve. This means that significantly smaller parameters can be used in ECC than in other competitive systems such RSA and DSA, but with equivalent levels of security. Some benefits of having smaller key sizes include faster computations, and reductions in processing power, storage space and bandwidth. This makes ECC ideal for constrained environments such as pagers, PDAs, cellular phones and smart cards. The implementation of ECC, on the other hand, requires several choices such as the type of the underlying finite field, algorithms for implementing the finite field arithmetic and so on. In this paper we give we presen an selective overview of the main methods.
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.
Faster scalar multiplication on Koblitz curves combining point halving with the Frobenius endomorphism
 in Proceedings of the 7th International Workshop on Theory and Practice in Public Key Cryptography, PKC 2004
"... on occasion of the birth of his daughter Seraina. Abstract. Let E be an elliptic curve defined over F2n. The inverse operation of point doubling, called point halving, can be done up to three times as fast as doubling. Some authors have therefore proposed to perform a scalar multiplication by an “ha ..."
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Cited by 16 (9 self)
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on occasion of the birth of his daughter Seraina. Abstract. Let E be an elliptic curve defined over F2n. The inverse operation of point doubling, called point halving, can be done up to three times as fast as doubling. Some authors have therefore proposed to perform a scalar multiplication by an “halveandadd ” algorithm, which is faster than the classical doubleandadd method. If the coefficients of the equation defining the curve lie in a small subfield of F2n, one can use the Frobenius endomorphism τ of the field extension to replace doublings. Since the cost of τ is negligible if normal bases are used, the scalar multiplication is written in “base τ ” and the resulting “τandadd ” algorithm gives very good performance. For elliptic Koblitz curves, this work combines the two ideas for the first time to achieve a novel decomposition of the scalar. This gives a new scalar multiplication algorithm which is up to 14.29 % faster than the Frobenius method, without any additional precomputation.
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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Cited by 9 (0 self)
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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...
Improved Algorithms for Arithmetic on Anomalous Binary Curves
 In Advances in Cryptography, Crypto '97
, 1997
"... . It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the elliptic ..."
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Cited by 7 (0 self)
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. It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. The basic operation is scalar multiplication: taking a given integer multiple of a given point on the curve. The cost of the protocols depends on that of the elliptic scalar multiplication operation. Koblitz introduced a family of curves which admit especially fast elliptic scalar multiplication. His algorithm was later modified by Meier and Staffelbach. We give an improved version of the algorithm which runs 50% faster than any previous version. It is based on a new kind of representation of an integer, analogous to certain kinds of binary expansions. We also outline further speedups using precomputation and storage. Keywords: elliptic curves, exponentiation, publickey cryptography. 1 Introduction It has become increasingly common to implement discretelogarithm based publickey protocols on elliptic curves over finite fields. More precisely, one work...
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
Open Problems And Conjectures In Finite Fields
 In Finite Fields and Applications (S.D. Cohen and H. Niederreiter Eds.), Cambridge University Press, Lecture Note Series of London Math. Society
, 1996
"... We discuss a number of open problems and conjectures in the theory and application of finite fields. We also provide a brief discussion of the status as well as references related to each problem. 1 Introduction In this paper we try to summarise some interesting and/or important questions in the th ..."
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Cited by 4 (0 self)
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We discuss a number of open problems and conjectures in the theory and application of finite fields. We also provide a brief discussion of the status as well as references related to each problem. 1 Introduction In this paper we try to summarise some interesting and/or important questions in the theory and application of finite fields. These questions obviously reflect our personal tastes but we have indeed tried to consider questions of general interest. We hope that these questions and even more, the methods developed for their solutions, will be of interest to other researchers. The reader may wonder why some questions we call `Problems' and some we call `Conjectures'. Roughly speaking, we use the term Conjecture if we (and very often others) believe the statement to be true while the term Problem is used to indicate all other statements. In general we feel that our conjectures may be more difficult to resolve than our problems. The standard notions of the theory of finite fields w...