Building cryptographic schemes based on elliptic curves over rational numbers

Abstract

The possibility of using elliptic curves over the rational field of non-zero ranks in cryptographic schemes is studied. For the first time, the construction of cryptosystems is proposed the security of which is based on the complexity of solving the knapsack problem on elliptic curves over rational numbers of non-zero ranks. A new approach to the use of elliptic curves for cryptographic schemes is proposed. A few experiments have been carried out to estimate the heights characteristic of points on elliptic curves of infinite order. A model of a cryptosystem resistant to computations on a quantum computer and based on rational points of an infinite order curve is proposed. A study of the security and effectiveness of the proposed scheme has been carried out. An attack on the secret search in such a cryptosystem is implemented and it is shown that the complexity of the attack is exponential. The proposed solution can be applied in the construction of real cryptographic schemes as well as cryptographic protocols.

Keywords

• elliptic curves
• rational numbers
• curve rank
• asymmetric encryption
• knapsack problem