‎Oblivious transfer is one of the important tools in cryptography‎, ‎in which a sender sends a message to a receiver with a probability between 0 and 1‎, ‎while the sender remains oblivious that the receiver has received the message‎.‎A flavor of $OT$ schemes is chosen $t$-out-of-$k$ oblivious transfer ($OT^t_k$)‎. ‎In an $OT^t_k$ scheme‎, ‎a sender transfers $k$ messages to a receiver‎, ‎the receiver can learn only $t$ of them‎, ‎and the sender remains oblivious to which secrets are extracted by the receiver‎. ‎In this paper‎, ‎we first propose a type of Diffie-Hellman key exchange protocol using the generalized Jacobian of elliptic curves‎. ‎Next‎, ‎we introduce simple‎, ‎secure two-round algorithms for $OT$‎, ‎$OT^1_2$‎, ‎$OT^t_k$‎.‎The security of proposed protocols is based on the intractability assumption of solving discrete logarithm problem; furthermore‎, ‎in our $OT$ schemes‎, ‎it is not necessary to map the messages to the points on the elliptic curve‎.

Many researchers have explored the construction of isogenies between particular elliptic curves, largely due to the extensive range of their practical applications. In this paper, we use the relationship between supersingular elliptic curves over $\mathbb{F}_{p}$ and binary quadratic forms to generate isogenies connecting supersingular elliptic curves over $\mathbb{F}_{p}$.

Let E be an elliptic curve over the finite field Fq, P a point in E(Fq) of order n, and Q a point in the group generated by P. The discrete logarithm problem on E is to find the number k such that Q = kP. In this paper we reduce the discrete logarithm problem on E[n] to the discrete logarithm on the group F*q , the multiplicative group of nonzero elements of Fq, in the case where n ï q - 1, using generalized jacobian of E.

In this article, we construct families of elliptic curves arising from the Heron triangles and Diophantine triples with the Mordell-Weil torsion subgroup of Z/2Z´Z/2Z. These families have ranks at least 2 and 3, respectively, and contain particular examples with rank equal to 7.

This paper provides an overview of elliptic curves and their use in cryptography. The focus of the paper is on the performance advantages obtained in the wireless environments by using elliptic curve cryptography instead of traditional cryptosystems such as RSA. Specific applications to secure messaging and identity-based encryption are also discussed.