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Curve25519: new DiffieHellman speed records
 In Public Key Cryptography (PKC), SpringerVerlag LNCS 3958
, 2006
"... Abstract. This paper explains the design and implementation of a highsecurity ellipticcurveDiffieHellman function achieving recordsetting speeds: e.g., 832457 Pentium III cycles (with several side benefits: free key compression, free key validation, and stateoftheart timingattack protection) ..."
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Cited by 67 (22 self)
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Abstract. This paper explains the design and implementation of a highsecurity ellipticcurveDiffieHellman function achieving recordsetting speeds: e.g., 832457 Pentium III cycles (with several side benefits: free key compression, free key validation, and stateoftheart timingattack protection), more than twice as fast as other authors ’ results at the same conjectured security level (with or without the side benefits). 1
Counting the Number of Points on Elliptic Curves Over Finite Fields: Strategies and Performances
, 1995
"... Cryptographic schemes using elliptic curves over finite fields require the computation of the cardinality of the curves. Dramatic progress have been achieved recently in that field by various authors. The aim of this article is to highlight part of these improvements and to describe an efficient imp ..."
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Cited by 36 (6 self)
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Cryptographic schemes using elliptic curves over finite fields require the computation of the cardinality of the curves. Dramatic progress have been achieved recently in that field by various authors. The aim of this article is to highlight part of these improvements and to describe an efficient implementation of them in the particular case of the fields GF (2 n ), for n 600. 1 Introduction Elliptic curves have been used successfully to factor integers [26, 36], and prove the primality of large integers [6, 15, 4]. Moreover they turned out to be an interesting alternative to the use of Z=NZ in cryptographical schemes [33, 21]. Elliptic curve cryptosystems over finite fields have been built, see [5, 30]; some have been proposed in Z=NZ, N composite [23, 12, 42]. More applications were studied in [19, 22]. The interested reader should also consult [31]. In order to perform key exchange algorithms using an elliptic curve E over a finite field K, the cardinality of E must be known. Th...
Constructing Elliptic Curve Cryptosystems in Characteristic 2
, 1998
"... Since the group of an elliptic curve defined over a finite field F_q... The purpose of this paper is to describe how one can search for suitable elliptic curves with random coefficients using Schoof's algorithm. We treat the important special case of characteristic 2, where one has certain simp ..."
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Cited by 18 (1 self)
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Since the group of an elliptic curve defined over a finite field F_q... The purpose of this paper is to describe how one can search for suitable elliptic curves with random coefficients using Schoof's algorithm. We treat the important special case of characteristic 2, where one has certain simplifications in some of the algorithms.
Finding Good Random Elliptic Curves for Cryptosystems Defined over ...
 Advances in Cryptology { EUROCRYPT '97
, 1997
"... . One of the main difficulties for implementing cryptographic schemes based on elliptic curves defined over finite fields is the necessary computation of the cardinality of these curves. In the case of finite fields IF2 n , recent theoretical breakthroughs yield a significant speed up of the comput ..."
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. One of the main difficulties for implementing cryptographic schemes based on elliptic curves defined over finite fields is the necessary computation of the cardinality of these curves. In the case of finite fields IF2 n , recent theoretical breakthroughs yield a significant speed up of the computations. Once described some of these ideas in the first part of this paper, we show that our current implementation runs from 2 up to 10 times faster than what was done previously. In the second part, we exhibit a slight change of Schoof's algorithm to choose curves with a number of points "nearly" prime and so construct cryptosystems based on random elliptic curves instead of specific curves as it used to be. 1 Introduction It is well known that the discrete logarithm problem is hard on elliptic curves defined over finite fields IF q . This is due to the fact that the only known attacks (baby steps giant steps [Sha71], Pollard ae [Pol78] and PohligHellman [PH78] methods) are still exponen...
Fast BitLevel, WordLevel and Parallel Arithmetic in Finite Fields for Elliptic Curve Cryptosystems
, 1998
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The Sea Algorithm in Characteristic 2
"... The SchoofElkiesAtkin algorithm counts the number of rational points on elliptic curves over finite fields. This paper presents a number of optimizations specific for the characteristic two case. We give a detailed description of the computation of modular polynomials over F 2 n and the search ..."
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The SchoofElkiesAtkin algorithm counts the number of rational points on elliptic curves over finite fields. This paper presents a number of optimizations specific for the characteristic two case. We give a detailed description of the computation of modular polynomials over F 2 n and the search for an eigenvalue in the Elkies algorithm. With our implementation, we were able to count the number of rational points on a curve defined over F 2 1999 , which is the current world record for the characteristic two case.