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Analysis of the Xedni calculus attack
 Design, Codes and Cryptography
, 2000
"... Abstract. The xedni calculus attack on the elliptic curve discrete logarithm problem (ECDLP) involves lifting points from the finite field Fp to the rational numbers Q and then constructing an elliptic curve over Q that passes through them. If the lifted points are linearly dependent, then the ECDLP ..."
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Abstract. The xedni calculus attack on the elliptic curve discrete logarithm problem (ECDLP) involves lifting points from the finite field Fp to the rational numbers Q and then constructing an elliptic curve over Q that passes through them. If the lifted points are linearly dependent, then the ECDLP is solved. Our purpose is to analyze the practicality of this algorithm. We find that asymptotically the algorithm is virtually certain to fail, because of an absolute bound on the size of the coefficients of a relation satisfied by the lifted points. Moreover, even for smaller values of p experiments show that the odds against finding a suitable lifting are prohibitively high.
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of ..."
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Cited by 6 (2 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
Some remarks on Heegner point computations
, 2004
"... We make some remarks concerning Heegner point computations. One of our goals shall be to give an algorithm to find a nontorsion rational point on a given rank 1 elliptic curve. Much of this is taken from a section in Henri Cohen’s latest ..."
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We make some remarks concerning Heegner point computations. One of our goals shall be to give an algorithm to find a nontorsion rational point on a given rank 1 elliptic curve. Much of this is taken from a section in Henri Cohen’s latest
Curves Dy 2 = x 3 − x of odd analytic rank
, 2002
"... Abstract. For nonzero rational D, which may be taken to be a squarefree integer, let ED be the elliptic curve Dy 2 = x 3 − x over Q arising in the “congruent number ” problem. 1 It is known that the Lfunction of ED has sign −1, and thus odd analytic rank ran(ED), if and only if D is congruent to ..."
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Abstract. For nonzero rational D, which may be taken to be a squarefree integer, let ED be the elliptic curve Dy 2 = x 3 − x over Q arising in the “congruent number ” problem. 1 It is known that the Lfunction of ED has sign −1, and thus odd analytic rank ran(ED), if and only if D is congruent to 5, 6, or 7 mod 8. For such D, we expect by the conjecture of Birch and SwinnertonDyer that the arithmetic rank of each of these curves ED is odd, and therefore positive. We prove that ED has positive rank for each D such that D  is in one of the above congruence classes mod 8 and also satisfies D  < 10 6. Our proof is computational: we use the modular parametrization of E1 or E2 to construct a rational point PD on each ED from CM points on modular curves, and compute PD to enough accuracy to usually distinguish it from any of the rational torsion points on ED. In the 1375 cases in which we cannot numerically distinguish PD from (ED)tors, we surmise that PD is in fact a torsion point but that ED has rank 3, and prove that the rank is positive by searching for and finding a nontorsion rational point. We also report on the conjectural extension to D  < 10 7 of the list of curves ED with odd ran(ED)> 1, which raises several new questions.
From Global Class Fields . . . ShimuraTaniyamaWeil, BirchSwinnertonDyer and Riemann
, 2006
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HIGH RANK ELLIPTIC CURVES WITH PRESCRIBED TORSION GROUP OVER QUADRATIC FIELDS
"... Abstract. There are 26 possibilities for the torsion group of elliptic curves de ned over quadratic number elds. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible tors ..."
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Abstract. There are 26 possibilities for the torsion group of elliptic curves de ned over quadratic number elds. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic eld with this torsion group and with rank ≥ 2. 1.