Narrow Results

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p^∞-Selmer group of E over any strongly admissible p-adic Lie extension K_∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K_∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

Recently, by changing security requirements of computer networks, many public key schemes are introduced. One major shortcoming of identity-based cryptosystems is key screw. Certificateless public key cryptosystems were introduced to solve this problem. In this paper, a certificateless, public-key, multiple-key-agreement scheme will be offered which has some significant security properties such as perfect forward secrecy, strong security, and zero-knowledge proof. This scheme produces far more shared hidden keys per session in comparison with many existing schemes. In this paper, the security and the efficiency of the proposed scheme will be compared with some well-known current schemes.