Efficient Implementation of Schoof’s Algorithm

Izu, Tetsuya; Kogure, Jun; Noro, Masayuki; Yokoyama, Keitaro

doi:10.1007/3-540-49649-1_7

Cited by 16 publications

(22 citation statements)

References 22 publications

(36 reference statements)

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“…We used the same settings on the complexity estimate function complex for Atkin/Elkies methods and the isogeny cycles method as in [6], and made a special table for the two-cycles method based on data of actual computations. Same as in [6], we did not use the original Schoof's method. For the experiment, we pre-computed modular polynomials for curves over GF (2 n ).…”

Section: Resultsmentioning

confidence: 99%

“…In this section, we recall the SEA algorithm and give brief overview of the ICS [6]. Here we consider an elliptic curve E over a finite field GF (q) of q elements.…”

Section: Sea Algorithm and Icsmentioning

confidence: 99%

“…In [6], this "combination problem" was dealt with and the ICS was proposed as a systematic and practical answer. It has the following functions corresponding to the problem.…”

Section: Icsmentioning

confidence: 99%

“…k+s by using g k,s , we can use a technique similar to the match-and-sort technique in the isogeny cycles method ( [6] …”

Section: When We Compute T Modmentioning

confidence: 99%

“…Thus, for an efficient implementation, it is important to find an efficient combination of such subprocedures. In [6] the Intelligent Choice System (ICS for short) was proposed to give an efficient combination of subprocedures in the first stage and was applied to counting the orders of curves over large prime fields.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Implementation of Schoof’s Algorithm in Case of Characteristic 2

Izu

Kogure

Yokoyama

2000

Public Key Cryptography

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Abstract. In order to choose a secure elliptic curve for Elliptic Curve Cryptosystems, it is necessary to count the order of a randomly selected elliptic curve. Schoof's algorithm and its variants by Elkies and Atkin are known as efficient methods to count the orders of elliptic curves. Intelligent Choice System(ICS) was proposed to combine these methods efficiently. In this paper, we propose an improvement on the ICS. Further, we propose several implementation techniques in the characteristic 2 case.

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Section: Resultsmentioning

confidence: 99%

“…In this section, we recall the SEA algorithm and give brief overview of the ICS [6]. Here we consider an elliptic curve E over a finite field GF (q) of q elements.…”

Section: Sea Algorithm and Icsmentioning

confidence: 99%

“…In [6], this "combination problem" was dealt with and the ICS was proposed as a systematic and practical answer. It has the following functions corresponding to the problem.…”

Section: Icsmentioning

confidence: 99%

“…k+s by using g k,s , we can use a technique similar to the match-and-sort technique in the isogeny cycles method ( [6] …”

Section: When We Compute T Modmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient Implementation of Schoof’s Algorithm in Case of Characteristic 2

Izu

Kogure

Yokoyama

2000

Public Key Cryptography

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Order counting of elliptic curves defined over finite fields of characteristic 2

Izu

Kogure

Noro

et al. 2001

Electron Comm Jpn Pt III

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SUMMARYAlgorithms for counting the order of the rational points group are indispensable for the random selection of secure elliptic curves used in cryptosystems. In the case of curves over fields having large characteristics, it is possible to use the SEA (SchoofElkiesAtkin) algorithm [3,5,16], but in the case of elliptic curves defined over finite fields of characteristic 2, the SEA algorithm can be applied only in a modified form. In the case of characteristic 2, the authors implemented the Lercier [10] and Couveignes [2] algorithms, and combined them with an improved Intelligent Choice System [7,8]

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Finding Secure Curves with the Satoh-FGH Algorithm and an Early-Abort Strategy

Fouquet

Gaudry

Harley³

2001

Lecture Notes in Computer Science

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The use of elliptic curves in cryptography relies on the ability to count the number of points on a given curve. Before 1999, the SEA algorithm was the only efficient method known for random curves. Then Satoh proposed a new algorithm based on the canonical p-adic lift of the curve for p ≥ 5. In an earlier paper, the authors extended Satoh's method to the case of characteristics two and three. This paper presents an implementation of the Satoh-FGH algorithm and its application to the problem of finding curves suitable for cryptography. By combining Satoh-FGH and an early-abort strategy based on SEA, we are able to find secure random curves in characteristic two in much less time than previously reported. In particular we can generate curves widely considered to be as secure as RSA-1024 in less than one minute each on a fast workstation.

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Efficient Implementation of Schoof’s Algorithm

Cited by 16 publications

References 22 publications

Efficient Implementation of Schoof’s Algorithm in Case of Characteristic 2

Efficient Implementation of Schoof’s Algorithm in Case of Characteristic 2

Order counting of elliptic curves defined over finite fields of characteristic 2

Finding Secure Curves with the Satoh-FGH Algorithm and an Early-Abort Strategy

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