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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.
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.
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.
Sign Change Fault Attacks on Elliptic Curve Cryptosystems
 Fault Diagnosis and Tolerance in Cryptography 2006 (FDTC ’06), volume 4236 of Lecture Notes in Computer Science
, 2004
"... We present a new type of fault attacks on elliptic curve scalar multiplications: Sign Change Attacks. These attacks exploit di#erent number representations as they are often employed in modern cryptographic applications. Previously, fault attacks on elliptic curves aimed to force a device to out ..."
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Cited by 14 (0 self)
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We present a new type of fault attacks on elliptic curve scalar multiplications: Sign Change Attacks. These attacks exploit di#erent number representations as they are often employed in modern cryptographic applications. Previously, fault attacks on elliptic curves aimed to force a device to output points which are on a cryptographically weak curve. Such attacks can easily be defended against. Our attack produces points which do not leave the curve and are not easily detected. The paper also presents a revised scalar multiplication algorithm that provably protects against Sign Change Attacks.
Compact Representation of Elliptic Curve Points Over
 GF(2 n ). Research Contribution to IEEE P1363
, 1998
"... . A method is described to represent points on elliptic curves over F 2 n , in the context of elliptic curve cryptosystems, using n bits. The method allows for full recovery of the x and y components of the point. This improves on the naive representation using 2n bits, and on the compressed represe ..."
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Cited by 11 (0 self)
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. A method is described to represent points on elliptic curves over F 2 n , in the context of elliptic curve cryptosystems, using n bits. The method allows for full recovery of the x and y components of the point. This improves on the naive representation using 2n bits, and on the compressed representation described in draft standard IEEE P1363, which uses n+1 bits. The representation described in this disclosure is optimal for the general case of a cryptosystem over F 2 n . Elliptic curve (EC) cryptography is gaining favor as an efficient and attractive alternative to the more conventional public key schemes, e.g., RSA. EC cryptosystems are based on operations involving points on an elliptic curve over a finite (or Galois) field. Popular choices for the underlying finite field are F p , a field of integers modulo p for a (very large) prime number p, and F 2 n , a finite field of characteristic two and dimension n. This disclosure focuses on the latter type of field. The following...
Design of elliptic curves with controllable lower boundary of extension degree for reduction attacks
 In Advances in Cryptology  CRYPTO '94
, 1994
"... Abstract. In this paper, we present a design strategy of elliptic curves whose extension degrees needed for reduction attacks have a controllable lower boundary, based on the complex multiplication fields method of Atkin and Morain over prime fields. 1 ..."
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Cited by 6 (0 self)
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Abstract. In this paper, we present a design strategy of elliptic curves whose extension degrees needed for reduction attacks have a controllable lower boundary, based on the complex multiplication fields method of Atkin and Morain over prime fields. 1
Specification For: Efficient implementation of Elliptic Curve Cryptosystems over binary Galois Fields, GF(2 m) in normal and polynomial bases
"... In this project, we will seek to efficiently implement an elliptic curve cryptographic library over a Galois field in normal and polynomial bases. Elliptic curve cryptosystems are particularly valuable as they are generally faster than other commonly used public key cryptosystems, and they require a ..."
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Cited by 2 (0 self)
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In this project, we will seek to efficiently implement an elliptic curve cryptographic library over a Galois field in normal and polynomial bases. Elliptic curve cryptosystems are particularly valuable as they are generally faster than other commonly used public key cryptosystems, and they require a smaller key size. Many performance studies have found that elliptic curve
Public Key Cryptosystems using Elliptic Curves
, 1997
"... This report is a survey on public key cryptosystems that use the theory of elliptic curves. A considerable part will be about the theory of elliptic curves. Encryption systems, digital signature schemes and key agreement schemes using elliptic curves will be described. Their workload and bandwidth w ..."
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Cited by 1 (0 self)
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This report is a survey on public key cryptosystems that use the theory of elliptic curves. A considerable part will be about the theory of elliptic curves. Encryption systems, digital signature schemes and key agreement schemes using elliptic curves will be described. Their workload and bandwidth will be addressed and some attacks will be described. For all systems the security is based either on the elliptic curve discrete logarithm problem or on the difficulty of factorization. The differences between conventional and elliptic curve systems shall be addressed. Systems based on the elliptic curve discrete logarithm problem can be used with shorter keys to provide the same security, compared to similar conventional systems. Elliptic curve systems based on factoring are slightly more resistant as conventional systems against some attacks.
Compact Representation of Elliptic Curve Points over
 GF(2 n ). Research Contribution to IEEE P1363
, 1998
"... A method is described to represent points on elliptic curves over F 2 n , in the context of elliptic curve cryptosystems, using n bits. The method allows for full recovery of the x and y components of the point. This improves on the naive representation using 2n bits, and on a previously known c ..."
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A method is described to represent points on elliptic curves over F 2 n , in the context of elliptic curve cryptosystems, using n bits. The method allows for full recovery of the x and y components of the point. This improves on the naive representation using 2n bits, and on a previously known compressed representation using n+ 1 bits. Since n bits are necessary to represent a point in the general case of a cryptosystem over F 2 n , the representation described in this note is minimal. Keywords: finite fields, elliptic curves, cryptography. 1 Background Elliptic curve (EC) cryptography is gaining favor as an efficient and attractive alternative to the more conventional public key schemes, e.g., RSA. EC cryptosystems are based on operations involving points on an elliptic curve over a finite field. Popular choices for the underlying finite field are F p , the integers modulo p for a (large) prime number p, and F 2 n , a finite field of characteristic two and dimension n. Thi...
Lecture Notes on Cryptography
, 1999
"... Contents 1 Preliminaries 1 1.1 Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Elliptic curve cryptography 3 2.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The size of an elliptic curve . . . . . . . . . . . . . . . . . . . . . 8 2.3 ..."
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Contents 1 Preliminaries 1 1.1 Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Elliptic curve cryptography 3 2.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The size of an elliptic curve . . . . . . . . . . . . . . . . . . . . . 8 2.3 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Discrete logarithms and Pollard's rho method . . . . . . . . . . . 13 2.5 Synopsis elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Pseudorandom generators 17 3.1 Pseudorandom generators . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Distinguishers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 The NisanWigderson generator . . . . . . . . . . . . . . . . . . . 26 3.5 Construction of good designs . . . . . . . . . . . . . . . . . . . . . 29 3.6 Deter