‎In this paper‎, ‎we propose some Diffie-Hellman type key exchange protocols using isogenies of elliptic curves‎. ‎The first method which uses the endomorphism ring of an ordinary elliptic curve $ E $‎, ‎is a straightforward generalization of elliptic curve Diffie-Hellman key exchange‎. ‎The method uses commutativity of the endomorphism ring $ End(E) $‎. ‎Then using dual isogenies‎, ‎we propose...

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...

in this paper the family of elliptic curves over q given by the equation ep :y2 = (x - p)3 + x3 + (x + p)3 where p is a prime number, is studied. itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(ep(q)) = 0 or rank(ep(q)) = 1 orrank(ep(q))≥2 are given.

In recent years it has been trying that with regard to the question of computational complexity of discrete logarithm more strength and less in the elliptic curve than other hard issues, applications such as elliptic curve cryptography, a blind  digital signature method, other methods such as encryption replacement DLP. In this paper, a new blind digital signature scheme based on elliptic curve...

Elliptic Curve Cryptosystems (ECC) have recently received significant attention by researchers due to their high performance such as low computational cost and small key size. In this paper a novel untraceable blind signature scheme is presented. Since the security of proposed method is based on difficulty of solving discrete logarithm over an elliptic curve, performance of the proposed scheme ...

‎By the Mordell‎- ‎Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎This paper studies the rank of the family Epq:y2=x3-pqx of elliptic curves‎, ‎where p and q are distinct primes‎. ‎We give infinite families of elliptic curves of the form y2=x3-pqx with rank two‎, ‎three and four‎, ‎assuming a conjecture of Schinzel ...

 The Mordell-Weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎In our previous paper, H‎. ‎Daghigh‎, ‎and S‎. ‎Didari‎, On the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎Bull‎. ‎Iranian Math‎. ‎Soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using Selmer groups‎, ‎we have shown that for a prime $p...

the mordell-weil theorem states that the group of rational points‎ ‎on an elliptic curve over the rational numbers is a finitely‎ ‎generated abelian group‎. ‎in our previous paper, h‎. ‎daghigh‎, ‎and s‎. ‎didari‎, on the elliptic curves of the form $ y^2=x^3-3px$‎, ‎‎bull‎. ‎iranian math‎. ‎soc‎.‎‎ 40 (2014)‎, no‎. ‎5‎, ‎1119--1133‎.‎, ‎using selmer groups‎, ‎we have shown that for a prime $p$...

‎It is shown that the knowledge of a surjective morphism $Xto Y$ of complex‎ ‎curves can be effectively used‎ ‎to make explicit calculations‎. ‎The method is demonstrated‎ ‎by the calculation of $j(ntau)$ (for some small $n$) in terms of $j(tau)$ for the elliptic curve ‎with period lattice $(1,tau)$‎, ‎the period matrix for the Jacobian of a family of genus-$2$ curves‎ ‎complementing the classi...

Elliptic curves in Hesse form admit more suitable arithmetic than ones in Weierstrass form. But elliptic curve cryptosystems usually use Weierstrass form. It is known that both those forms are birationally equivalent. Birational equivalence is relatively hard to compute. We prove that elliptic curves in Hesse form and in Weierstrass form are linearly equivalent over initial field or its small e...