Nonlinearities in Elliptic Curve Authentication

Ramzi Alsaedi,Nicolae Constantinescu,Vicenţiu Rādulescu

doi:10.3390/e16095144

Ramzi Alsaedi, Nicolae Constantinescu + Show 1 more

Open Access

https://doi.org/10.3390/e16095144

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Abstract

In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area means finding the majority of points on the elliptic curve and minimizing the time to compute the solution in contrast with the necessary time to compute the inverse solution. We can compute the positive solution of PDE (partial differential equation) like oscillations of f(s)/s around the principal eigenvalue λ1 of -Δ in H 0 1 (Ω).Translating mathematics into cryptographic applications will be relevant in everyday life, where in there are situations in which two parts that communicate need a third part to confirm this process. For example, if two persons want to agree on something they need an impartial person to confirm this agreement, like a notary. This third part does not influence in anyway the communication process. It is just a witness to the agreement. We present a system where the communicating parties do not authenticate one another. Each party authenticates itself to a third part who also sends the keys for the encryption/decryption process. Another advantage of such a system is that if someone (sender) wants to transmit messages to more than one person (receivers), he needs only one authentication, unlike the classic systems where he would need to authenticate himself to each receiver. We propose an authentication method based on zero-knowledge and elliptic curves.

Highlights

The system we propose has three components: two parties that communicate and one party that authenticates them and provides the keys for the cryptosystem used
According with [9], from all points which define an elliptic curve, only a part can be used on applications, we can found the special points with properties in this way, called frontier points: (1) |E(Fp )| = c · l where l > 2160 a prime and c a positive integer. |E(Fp )| denotes the cardinal of the set of points on E over Fp
We propose a zero-knowledge authentication based on elliptic curves and on the algorithms described in the previous section

Summary

Introduction

The system we propose has three components: two parties that communicate and one party that authenticates them and provides the keys for the cryptosystem used. The first password authentication protocol used on a network proven secure was presented by Halevi and Krawczyk [1] Their protocol prevents leakage of information and the server’s private key can be verified by the user. A strategy S is zero-knowledge on the set A if for any feasible strategy B exists a feasible computation C so that the following are computationally indistinguishable: From this definition, any information obtained by interacting with S on some input a, can be obtained from a without interacting with S [6]. The verifier knows the right answer before communicating with the prover He cannot possibly obtain any new information. A passive adversary cannot obtain new information from the prover

Mathematical Preliminaries

Frontier Points on Elliptic Curves

Nonliniarities on Elliptic Curves

Counting the Elliptic Curve’s Frontier Points

Our Method

Conclusions