Псевдовипадкові криптостійкі послідовності і подільність точки скрученої кривої Едвардса навпіл над простими і складеними полями

Estimates of the complexity of the point division operation into two for twisted Edwards curve are obtained in comparison with the doubling of the point. One of the applications of the divisibility properties of a point into two is considered to determine the order of a point in a cryptosystem. The cryptological security of the pseudo-random sequence generator proposed by the author is shown on the basis of a curve in the form of Edwards. A new generation scheme and a new one-sided function of a pseudo-random cryptological security sequence based on these curves are proposed. The degree of embedding of these curves into a finite field for pairing on friendly elliptic curves of prime order or almost prime order is investigated. Pairingfriendly curves of prime or near-prime order are absolutely essential in certain pairing-based schemes like short signatures with longer useful life. For this goal we construct friendly curves on base of family of twisted Edwards curves. The possibility of constructing a twisted Edwards order curve, that is, one that has a minimal cofactor 4, has been found. A solution for the inverse doubling problem is obtained for quasi-elliptic curves that represented in the twisted Edwards form. Also its application to the proving of cryptographic pseudo-random sequence generator. It makes it possible to prove the cryptological security of the pseudo-random sequence we developed.