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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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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...
Design of Hyperelliptic Cryptosystems in small Characteristic and a Software Implementation over F2 n
 In Advances in Cryptology  ASIACRYPT '98
, 1998
"... Abstract. We investigate the discrete logarithm problem over jacobians of hyperelliptic curves suitable for publickey cryptosystems. We focus on the case when the definition field has small characteristic 2, 3, 5 and 7, then we present hyperelliptic cryptosystems that resist against all known attac ..."
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Abstract. We investigate the discrete logarithm problem over jacobians of hyperelliptic curves suitable for publickey cryptosystems. We focus on the case when the definition field has small characteristic 2, 3, 5 and 7, then we present hyperelliptic cryptosystems that resist against all known attacks. We further implement our designed hyperelliptic cryptosystems over finite fields F2n in software on Alpha and PentiumII computers. Our results indicate that if we choose curves carefully, hyperelliptic cryptosystems do have practical performance.
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...
Efficient Implementation of Schoof's Algorithm
 Advances in Cryptology { ASIACRYPT '98
, 1999
"... . Schoof's algorithm is used to find a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as AtkinElkies's method, the ..."
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. Schoof's algorithm is used to find a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as AtkinElkies's method, the isogeny cycles method, and trial search by matchandsort techniques, we can count the number of rational points on an elliptic curve over GF (p) in a reasonable time, where p is a prime whose size is around 240bits. 1 Introduction When we use the elliptic curve cryptosystem [9, 17] (ECC for short), we first have to define an elliptic curve over a finite field. Then, all cryptographic operations will be performed on the group of rational points on the curve. Since all the curves are not necessarily secure, we should be very careful when we choose an elliptic curve for ECC. There are several methods to select a curve for ECC, such as Schoof's method [22], CM(Complex Multiplication) method [2, 18, 10,...
Elliptic Curve Cryptosystems
, 1997
"... The application of elliptic curves to the field of cryptography has been relatively recent. It has opened up a wealth of possibilities in terms of security, encryption, and realworld applications. In particular, we are interested in publickey cryptosystems that use the elliptic curve discrete loga ..."
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The application of elliptic curves to the field of cryptography has been relatively recent. It has opened up a wealth of possibilities in terms of security, encryption, and realworld applications. In particular, we are interested in publickey cryptosystems that use the elliptic curve discrete logarithm problem to establish security. The objective of this thesis is to assemble the most important facts and findings into a broad, unified overview of this field. To illustrate certain points, we also discuss a sample implementation of the elliptic curve analogue of the El Gamal cryptosystem.
Counting points on elliptic curves over F p n using Couveignes's algorithm
, 1995
"... The heart of the improvements of Elkies to Schoof's algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies' approach was well suited for the case where the characteristic of the field is large. Couveignes sh ..."
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The heart of the improvements of Elkies to Schoof's algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies' approach was well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes's algorithm and to give numerous computational examples. In particular, we describe the use of fast algorithms for performing incremental operations on series. We will also insist on the particular case of the characteristic 2. 1 Introduction Elliptic curves have been used successfully to factor integers [25, 34], and prove the primality of large integers [4, 18, 3]. Moreover they turned out to be an interesting alternative to the use of Z=NZin cryptographical schemes. The first schemes were presented in [33, 23] and followed by many more (see for instance [31...
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.