Elliptic Curves over Number Fields
Heights on elliptic curves over number fields, period lattices, and complex elliptic logarithms
... a number of approximation techniques which eventually yield an approximate region corresponding to each inequality, where finding a solution to the system of these inequalities is equivalent to finding the ...
233
Darmon points on elliptic curves over number fields of arbitrary signature
... on elliptic curves defined over arbitrary number fields F ...work over any pair (E/F, K/F ), under the minimal assumptions that E/F is modular and sign(E/K) = − 1, which are ...
36
A criterion to rule out torsion groups for elliptic curves over number fields
... any elliptic curve over suitable (families of) number ...of elliptic curves over cubic and quartic ...of elliptic curves occurring over a specific cubic ...
13
On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields
... of elliptic curves having good reduction everywhere over certain real quad- ratic fields m for ...groups over some real quadratic fields via ...cubic fields. ...
8
Universal adelic groups for imaginary quadratic number fields and elliptic curves
... a number field K, the first invariants we defined were the ring of integers O K , its unit group O ∗ K , and its class group Cl K ...two number fields have isomorphic rings of integers, then they are ...
114
Modular Elliptic Curves over Quartic CM Fields
... given elliptic curve ...defined over number fields, where we do not know whether all elliptic curves are modular, but where there exist techniques to prove isomorphism of Galois ...
186
On elliptic curves of prime power conductor over imaginary quadratic fields with class number one
... Such curves have discriminant 2 8 uε 2 π r , so odd ...other number fields of class number one, as the number of units modulo squares is always ...quadratic fields is of ...
34
Torsion of elliptic curves over real quadratic fields of smallest discriminant
... of all K-rational points on E, together with the base point O, is an abelian group. He also conjectured that this group is finitely generated when K is the field Q of rational numbers. In 1922, Mordell proved this ...
9
Efficient Arithmetic on Elliptic Curves over Fields of Characteristic Three
... We compared the performance of the proposed formulae to the previously best results for different coordinates systems. It is shown that the new formulae are superior to the previously known ones. It should be pointed out ...
14
Fast Algorithms for Arithmetic on Elliptic Curves Over Prime Fields
... I would like to thank my supervisors, Drs. Renate Scheidler and Michael J. Jacob- son, who have done more for me than I could have asked for. I would also like to acknowledge the glorious invention of cheese and all the ...
232
A Note on Point Multiplication on Supersingular Elliptic Curves over Ternary Fields
... The point multiplication is performed by curve arithmetic operations such as point addition, doubling and tripling which in turn are performed by field arithmetic operations such as addition, subtraction, multiplication, ...
7
On the Static Diffie-Hellman Problem on Elliptic Curves over Extension Fields
... the number of terms, and substituted x R into this partially symmmetrised version of f 6 ...for curves over these fields to be impractical given our resources at the present ...base ...
20
Number of Jacobi quartic curves over finite fields
... Introduction Elliptic curve cryptosystems were proposed by Miller (1986) and by Koblitz (1987) which relies on the difficulty of the elliptic curve discrete logarithmic ...on elliptic curves ...
11
Multiplication in Finite Fields and Elliptic Curves
... of elliptic curve defined over F 2 n and F 3 n ...a number of cores, it is thus interesting to adapt the implementation of a scalar multiplication in order to execute it in parallel on these multiple ...
67
Faster arithmetic on elliptic curves using Fp2. Application to GLV-GLS and NIST elliptic curves over Fp isomorphic to twisted Hessian curves over fields extension
... the number of processor cycles required for multiplication in F p ...the number of processor cycles for other ...of elliptic curve in short Weierstrass form over F h where h is about twice ...
22
A reduction of the space for the parallelized Pollard lambda search on elliptic curves over prime finite fields and on anomalous binary elliptic curves
... |R| = |x|. We assume that the integer numbers e, a 1 , a 2 , a 3 , a 4 take O(log |x|) bits for their representation. First we consider if there exists a natural number n > 1 and a point R 1 ∈ E(Q) such that nR 1 ...
11
On Serre's uniformity conjecture for semistable elliptic curves over totally real fields
... semistable elliptic curve without complex multiplication, then the representation ρ E,p is surjective for any prime p ≥ ...class number of Q is 1, we have ψ = ...contradiction. Over a number ...
8
New Fast Algorithms for Arithmetic on Elliptic Curves over Fields of Characteristic Three
... Table 6. Proposed Hessian point doubling algorithm ▍ Note 3. In the above point addition algorithm D − 1 is a constant which can be precomputed from the curve equation and the last multiplication is by the constant. In ...
13
Improving the Complexity of Index Calculus Algorithms in Elliptic Curves over Binary Fields
... the number of equations minus the number of ...this number is > 0, and under a reasonable linear independence assumption confirmed by our experimental results, the computation is ...
18