Elliptic Curve Cryptography Parameterization for Password Security
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Existing password-authenticated key exchange protocols using elliptic curve cryptography are vulnerable to attacks that can deduce the password due to redundancy in encrypted exchanges, allowing attackers to test passwords and potentially uncover the password.
Innovation Solution
A method employing a surjective function F that maps input parameters to points on an elliptic curve in a way that any input pair (k, k') results in a point on the curve, regardless of whether they satisfy the classical elliptic curve equation, ensuring that the encrypted parameters do not reveal information about the password.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If random values are exchanged in encrypted form using classical elliptic curve representation, then password authentication can be performed, but redundancy of information allows attackers to deduce the password
Solution Approach 1:
The patent changes the parameter representation of elliptic curve points by introducing a surjective function F that maps arbitrary field elements to curve points. Instead of using standard (x,y) coordinates that satisfy the curve equation, the patent uses parameters (k,k') that are transformed via F to produce valid curve points. This parameter transformation eliminates the redundancy that attackers exploit, as the encrypted exchanges no longer contain information that can be used to verify password correctness.
Solution Approach 2:
The surjective function F acts as an intermediary between the random values exchanged during authentication and the elliptic curve points. This intermediary function masks the relationship between the exchanged parameters and the actual curve points, preventing attackers from directly analyzing the encrypted communications to deduce password information while still enabling valid authentication.
2Object-affected harmful factors
If a surjective function F is used to map parameters to elliptic curve points, then password security is enhanced by eliminating redundancy, but the mathematical complexity of the system increases
Solution Approach 1:
The patent introduces a surjective function F that transforms simple field elements into elliptic curve points through a defined mathematical relationship. This parameter change maintains security by eliminating the redundancy in classical representations while using systematic mathematical transformations that, although more complex than standard coordinates, follow clear algebraic rules that can be efficiently computed and verified.
Data Source
Figure 1~2
Figure 3
Figure 4
AI summary
The invention relates to the implementation of the control of a device (10) by a controller (11) on the basis of a password (p). For this purpose, the invention comprises determining (21), at the device or at the controller and on the basis of a random value r1, a point P(X,Y) on an elliptic curve in a finite body Fq, q being an integer, according to the equation: Ea, b (x, y) : x3 + ax + b = y2 (1). The invention then comprises obtaining (22) first and second parameters k and k', such that P(X,Y)=F(K,k'), where F is a surjective function of FqxFq, in Fq, and then obtaining (23) the first and second parameters in an encrypted format by encryption in accordance with the password. The invention finally comprises transmitting (24) the first and second encrypted parameters to the controller. During the control, the function F is used, such that, whatever the values of z and z', which are input elements of Fq, F(z,z') is a point on the elliptic curve and the input elements do not satisfy the equation (1).