Abelian varieties over finite fields

Waterhouse, William C.

doi:10.24033/asens.1183

Cited by 423 publications

(230 citation statements)

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“…Similar results for the case of elliptic curves over F 2 m can be deduced from the work of Waterhouse [122] and Schoof [106].…”

Section: Choosing a Curve At Randomsupporting

confidence: 68%

“…A result of Waterhouse [122] states that if q is a prime, then for each t satisfying |t| ≤ 2 √ q there exists at least one elliptic curve E defined over F q with #E(F q ) = q + 1 − t. If q is a power of 2, then for each even t satisfying |t| ≤ 2 √ q there exists at least one (non-supersingular) elliptic curve E defined over F q with #E(F q ) = q + 1 − t. Schoof [106] derived a formula for the number of isomorphism classes of elliptic curves defined over F q with #E(F q ) = q + 1 − t for each t satisfying |t| ≤ 2 √ q.…”

Section: Psfrag Replacementsmentioning

confidence: 99%

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A Survey of Public-Key Cryptosystems

Koblitz¹,

Menezes²

2004

SIAM Rev.

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“…Similar results for the case of elliptic curves over F 2 m can be deduced from the work of Waterhouse [122] and Schoof [106].…”

Section: Choosing a Curve At Randomsupporting

confidence: 68%

Section: Psfrag Replacementsmentioning

confidence: 99%

A Survey of Public-Key Cryptosystems

Koblitz¹,

Menezes²

2004

SIAM Rev.

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“…Proposition 3.5 (Deuring-Waterhouse [17]). -Les polynômes caractéristi-ques des courbes elliptiques supersingulières sur k sont…”

Section: Férard (é)unclassified

“…Proposition 3.6 (Waterhouse [17]). -Soit A une variété abélienne simple sur k. Soit h A le polynôme caractéristique de A sur k. Soit π une racine de h A dans C. Supposons que π a un conjugué réel.…”

Section: Férard (é)unclassified

Poids des duaux des codes BCH de distance prescrite $2^a+1$ et sommes exponentielles

Férard¹

2002

Bul. Soc. Math. France

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Résumé. -Soit n un entier pair. On considère un code BCH binaire Cn de longueur 2 n − 1 et de distance prescrite 2 a + 1 avec a ≥ 3. Le poids d'un mot non nul du dual de Cn peut s'exprimer en fonction d'une somme exponentielle. Nous montrerons que cette somme n'atteint pas la borne de Weil et nous proposerons une amélioration de celle-ci. En conséquence, nous obtiendrons une amélioration de la borne de CarlitzUchiyama sur le poids des mots du dual de Cn.Abstract (Weight of duals of BCH codes of designed distance 2 a + 1 and exponential sums)Let n be an even integer. We consider a binary BCH code Cn of length 2 n − 1 and designed distance 2 a + 1 with a ≥ 3. The weight of a nonzero codeword of the dual of Cn is linked to the value of an exponential sum. We will show that this exponential sum does not reach the Weil bound and we will improve this bound. Thus, we obtain an improvement of the Carlitz-Uchiyama bound on the weights of the words of the dual of Cn. IntroductionSoit n un entier strictement positif. Soit C n un code BCH binaire de longueur q − 1 = 2 n − 1 et de distance prescrite 2t + 1. Le poids w d'un mot de code non

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“…Given t an integer, \t\ < 2g 1 / 2 then WATERHOUSE [3] proved that there exists an elliptic curve over Fq with q + 1 -t rational points if and oniy if, writing q = j/ 1 , p prime, one of thé following conditions is satisfied :…”

mentioning

confidence: 99%

A note on elliptic curves over finite fields

Voloch¹

1988

Bul. Soc. Math. France

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Abelian varieties over finite fields

Cited by 423 publications

References 7 publications

A Survey of Public-Key Cryptosystems

A Survey of Public-Key Cryptosystems

Poids des duaux des codes BCH de distance prescrite $2^a+1$ et sommes exponentielles

A note on elliptic curves over finite fields

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