Oblivious Transfer Using Generalized Jacobian of Elliptic Curves

‎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.