An elliptic curve analogue of Pillai’s lower bound on primitive roots

On the elliptic curve analogue of the sum-product problem

Let E be an elliptic curve over a finite field Fq of q elements and x(P ) to denote the x-coordinate of a point P = (x(P ), y(P )) ∈ E. Let ⊕ denote the group operation in the Abelian group E(Fq) of Fq -rational points on E. We show that for any sets R,S ⊆ E(Fq) at least one of the sets { x(R)+ x(S): R ∈R, S ∈ S} and {x(R⊕ S): R ∈R, S ∈ S} is large. This question is motivated by a series of rec...

On an elliptic analogue of Zagier's conjecture.

Efficient elliptic curve cryptosystems

Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...

On simultaneous primitive roots

Given finitely many non zero rational numbers which are not ±1, we prove, under the assumption of Hypothesis H of Schinzel, necessary and sufficient conditions for the existence of infinitely many primes modulo which all the given numbers are simultaneously primitive roots. A stronger result where the density of the primes in consideration was computed was proved under the assumption of the Gen...

Explicit 4-descents on an Elliptic Curve

It is shown that the obvious method of descending from an element of the 2-Selmer group of an elliptic curve, E, will indeed give elements of order 1, 2 or 4 in the Weil-Chatelet group of E. Explicit algorithms for such a method are given.