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5,857
Identifying supersingular elliptic curves
- Preprint , 2011 (available from http://arxiv.org/abs/1107.1140
"... ar ..."
CM LIFTINGS OF SUPERSINGULAR ELLIPTIC CURVES
, 2009
"... Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by OD to supersingular elliptic curves in characteristic p. In the algorithm we first determine an e ..."
Abstract
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Cited by 1 (1 self)
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Assuming GRH, we present an algorithm which inputs a prime p and outputs the set of fundamental discriminants D < 0 such that the reduction map modulo a prime above p from elliptic curves with CM by OD to supersingular elliptic curves in characteristic p. In the algorithm we first determine
MINIMAL CM LIFTINGS OF SUPERSINGULAR ELLIPTIC CURVES
"... Abstract. In this paper, we give a ‘direct ’ construction of the endomorphism ring of supersingular elliptic curves over a prime field Fp from ‘ideal classes ’ of Q( √−p). We use the result to prove that the result of Kaneko on ‘minimal ’ CM liftings of such supersingular elliptic curves is a best p ..."
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Abstract. In this paper, we give a ‘direct ’ construction of the endomorphism ring of supersingular elliptic curves over a prime field Fp from ‘ideal classes ’ of Q( √−p). We use the result to prove that the result of Kaneko on ‘minimal ’ CM liftings of such supersingular elliptic curves is a best
Finding More Non-Supersingular Elliptic Curves
"... Abstract—Finding suitable non-supersingular elliptic curves for pairing-based cryptosystems becomes an important issue for the modern public-key cryptography after the proposition of id-based encryption scheme and short signature scheme. In previous work different algorithms have been proposed for f ..."
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Abstract—Finding suitable non-supersingular elliptic curves for pairing-based cryptosystems becomes an important issue for the modern public-key cryptography after the proposition of id-based encryption scheme and short signature scheme. In previous work different algorithms have been proposed
ON SUPERSINGULAR ELLIPTIC CURVES AND HYPERGEOMETRIC FUNCTIONS
"... Abstract. The Legendre Family of elliptic curves has the remarkable property that ( both its periods and its supersingular locus have descriptions in terms of the ..."
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Abstract. The Legendre Family of elliptic curves has the remarkable property that ( both its periods and its supersingular locus have descriptions in terms of the
On a certain supersingular elliptic curve
, 2001
"... In this note, by means of determination of the number of rational points, it is shown that if F is a finite prime field of characteristic p satisfying p- 5 (mod 8) then the elliptic curve y2 =X (X 2+X + r) defined over F is supersingular where r = 1/8 E F. As an application, it is also shown that th ..."
Abstract
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Cited by 1 (1 self)
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In this note, by means of determination of the number of rational points, it is shown that if F is a finite prime field of characteristic p satisfying p- 5 (mod 8) then the elliptic curve y2 =X (X 2+X + r) defined over F is supersingular where r = 1/8 E F. As an application, it is also shown
SUPERSINGULAR ELLIPTIC CURVES AND MAXIMAL QUATERNIONIC ORDERS
"... Abstract. We give an explicit version of the “Deuring correspondence ” between supersingular elliptic curves and maximal quaternionic orders, by presenting a deterministic and explicit algorithm to compute it. In this talk all fields of positive characteristic will be either finite or function field ..."
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Abstract. We give an explicit version of the “Deuring correspondence ” between supersingular elliptic curves and maximal quaternionic orders, by presenting a deterministic and explicit algorithm to compute it. In this talk all fields of positive characteristic will be either finite or function
Explicit Generators for Endomorphism Rings of Supersingular Elliptic Curves
, 2004
"... Let p be a prime, and let E/Fp2 be a supersingular elliptic curve. Then it is straightforward to show that EndF̄p(E) is isomorphic to a maximal order within Qp,∞, the unique quaternion algebra ramified precisely at p and ∞. In fact, it is a well-known result of Deuring that this defines a 1: 1 corre ..."
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Let p be a prime, and let E/Fp2 be a supersingular elliptic curve. Then it is straightforward to show that EndF̄p(E) is isomorphic to a maximal order within Qp,∞, the unique quaternion algebra ramified precisely at p and ∞. In fact, it is a well-known result of Deuring that this defines a 1: 1
Evidence that XTR is more secure than supersingular elliptic curve cryptosystems
- J. Cryptology
, 2001
"... Abstract. We show that finding an efficiently computable injective homomorphism from the XTR subgroup into the group of points over GF(p 2) of a particular type of supersingular elliptic curve is at least as hard as solving the Diffie-Hellman problem in the XTR subgroup. This provides strong evidenc ..."
Abstract
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Cited by 97 (5 self)
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Abstract. We show that finding an efficiently computable injective homomorphism from the XTR subgroup into the group of points over GF(p 2) of a particular type of supersingular elliptic curve is at least as hard as solving the Diffie-Hellman problem in the XTR subgroup. This provides strong
• Constructing supersingular elliptic curves, Journal of Combinator...
"... Thesis: Constructing elliptic curves of prescribed order; advisor: P. Stevenhagen. ..."
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Cited by 7 (0 self)
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Thesis: Constructing elliptic curves of prescribed order; advisor: P. Stevenhagen.
Results 1 - 10
of
5,857
