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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.
A PublicKey Infrastructure for Key Distribution in TinyOS Based on Elliptic Curve Cryptography
, 2004
"... We present the first known implementation of elliptic curve cryptography over F2 p for sensor networks based on the 8bit, 7.3828MHz MICA2 mote. Through instrumentation of UC Berkeley's TinySec module, we argue that, although secretkey cryptography has been tractable in this domain for some time, ..."
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Cited by 182 (3 self)
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We present the first known implementation of elliptic curve cryptography over F2 p for sensor networks based on the 8bit, 7.3828MHz MICA2 mote. Through instrumentation of UC Berkeley's TinySec module, we argue that, although secretkey cryptography has been tractable in this domain for some time, there has remained a need for an efficient, secure mechanism for distribution of secret keys among nodes. Although publickey infrastructure has been thought impractical, we argue, through analysis of our own implementation for TinyOS of multiplication of points on elliptic curves, that publickey infrastructure is, in fact, viable for TinySec keys' distribution, even on the MICA2. We demonstrate that public keys can be generated within 34 seconds, and that shared secrets can be distributed among nodes in a sensor network within the same, using just over 1 kilobyte of SRAM and 34 kilobytes of ROM.
Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems
, 1999
"... Differential Power Analysis, first introduced by Kocher et al. in [14], is a powerful technique allowing to recover secret smart card information by monitoring power signals. In [14] a specific DPA attack against smartcards running the DES algorithm was described. As few as 1000 encryptions were su ..."
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Cited by 162 (2 self)
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Differential Power Analysis, first introduced by Kocher et al. in [14], is a powerful technique allowing to recover secret smart card information by monitoring power signals. In [14] a specific DPA attack against smartcards running the DES algorithm was described. As few as 1000 encryptions were sufficient to recover the secret key. In this paper we generalize DPA attack to elliptic curve (EC) cryptosystems and describe a DPA on EC DiffieHellman key exchange and EC ElGamal type encryption. Those attacks enable to recover the private key stored inside the smartcard. Moreover, we suggest countermeasures that thwart our attack.
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.
The Discrete Logarithm Problem On Elliptic Curves Of Trace One
 Journal of Cryptology
, 1999
"... In this short note we describe an elementary technique which leads to a linear algorithm for solving the discrete logarithm problem on elliptic curves of trace one. In practice the method described means that when choosing elliptic curves to use in cryptography one has to eliminate all curves who ..."
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Cited by 84 (2 self)
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In this short note we describe an elementary technique which leads to a linear algorithm for solving the discrete logarithm problem on elliptic curves of trace one. In practice the method described means that when choosing elliptic curves to use in cryptography one has to eliminate all curves whose group orders are equal to the order of the finite field.
An algorithm for solving the discrete log problem on hyperelliptic curves
, 2000
"... Abstract. We present an indexcalculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields. The complexity predicts that it is faster than the Rho method for genus greater than 4. To demonstrate the efficiency of our approach, we de ..."
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Cited by 78 (6 self)
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Abstract. We present an indexcalculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields. The complexity predicts that it is faster than the Rho method for genus greater than 4. To demonstrate the efficiency of our approach, we describe our breaking of a cryptosystem based on a curve of genus 6 recently proposed by Koblitz. 1
Efficient Algorithms for Elliptic Curve Cryptosystems
, 1997
"... Elliptic curves are the basis for a relative new class of publickey schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This th ..."
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Cited by 66 (9 self)
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Elliptic curves are the basis for a relative new class of publickey schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into lowlevel algorithms, which deal with arithmetic in the underlying finite field and highlevel algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the KaratsubaOfman Algorithm to multiplication in composite fields GF ((2 n ) m ). The second algorithm deals with efficient inversion in composite Galois fields of the form GF ((2 n ) m ). The third algorithm is an entirely new approach which accelerates the multiplication of points which i...
Optimal Extension Fields for Fast Arithmetic in PublicKey Algorithms
, 1998
"... Abstract. This contribution introduces a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of publickey cryptosystems based on elliptic and hyperelliptic curves. Whereas previous reported ..."
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Cited by 64 (13 self)
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Abstract. This contribution introduces a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of publickey cryptosystems based on elliptic and hyperelliptic curves. Whereas previous reported optimizations focus on finite fields of the form GF (p) and GF (2 m), an OEF is the class of fields GF (p m), for p a prime of special form and m a positive integer. Modern RISC workstation processors are optimized to perform integer arithmetic on integers of size up to the word size of the processor. Our construction employs wellknown techniques for fast finite field arithmetic which fully exploit the fast integer arithmetic found on these processors. In this paper, we describe our methods to perform the arithmetic in an OEF and the methods to construct OEFs. We provide a list of OEFs tailored for processors with 8, 16, 32, and 64 bit word sizes. We report on our application of this approach to construction of elliptic curve cryptosystems and demonstrate a substantial performance improvement over all previous reported software implementations of Galois field arithmetic for elliptic curves.
Faster Attacks on Elliptic Curve Cryptosystems
 Selected Areas in Cryptography, LNCS 1556
, 1998
"... The previously best attack known on elliptic curve cryptosystems used in practice was the parallel collision search based on Pollard's aemethod. The complexity of this attack is the square root of the prime order of the generating point used. For arbitrary curves, typically defined over GF (p) or G ..."
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Cited by 61 (1 self)
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The previously best attack known on elliptic curve cryptosystems used in practice was the parallel collision search based on Pollard's aemethod. The complexity of this attack is the square root of the prime order of the generating point used. For arbitrary curves, typically defined over GF (p) or GF (2 m ), the attack time can be reduced by a factor or p 2, a small improvement. For subfield curves, those defined over GF (2 ed ) with coefficients defining the curve restricted to GF (2 e ), the attack time can be reduced by a factor of p 2d. In particular for curves over GF (2 m ) with coefficients in GF (2), called anomalous binary curves or Koblitz curves, the attack time can be reduced by a factor of p 2m. These curves have structure which allows faster cryptosystem computations. Unfortunately, this structure also helps the attacker. In an example, the time required to compute an elliptic curve logarithm on an anomalous binary curve over GF (2 163 ) is reduced from 2 ...
Evaluation of discrete logarithms in a group of ptorsion points of an elliptic curve in characteristic p
 Mathematics of Computation
, 1998
"... Abstract. We show that to solve the discrete log problem in a subgroup of order p of an elliptic curve over the finite field of characteristic p one needs O(ln p) operations in this field. Let Fq be the finite field of q = p l elements. We define an elliptic curve E over Fq to be an equation of the ..."
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Cited by 58 (0 self)
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Abstract. We show that to solve the discrete log problem in a subgroup of order p of an elliptic curve over the finite field of characteristic p one needs O(ln p) operations in this field. Let Fq be the finite field of q = p l elements. We define an elliptic curve E over Fq to be an equation of the form y 2 = x 3 + Ax + B. We suppose p ̸ = 2,3. Let E(Fq) bethesetofpointsErational over Fq. Itisknown that Nq − q − 1  ≤2q 1/2 with Nq = E(Fq). The set E(Fq) is a finite abelian group with the “infinite point ” P ∞ as the identity element. The discrete logarithm problem is to compute an integer n such that Q = nP, where Q, P ∈ E(Fq), if such an n exists. This problem is of great significance in cryptology [1], [2]. Suppose that the point P generates a subgroup 〈P 〉 of order m. If (m, p) = 1, then the subgroup 〈P 〉 is isomorphic to some multiplicative subgroup of an extension F q k where q k ≡ 1(modm). The values of the isomorphism from 〈P 〉 to F ∗ q can be evaluated in a very simple manner. The complexity of the algorithm is