Deterministic Elliptic Curve Point Encoding for Constant-Time Cryptography
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Solution Overview
Problem
Existing elliptic curve-based cryptographic algorithms are probabilistic, making implementation time dependent on the message, which can reveal information to attackers, and attempts to mask this with constant-time processing are inefficient.
Innovation Solution
A deterministic method using the Euler function φ to obtain points on an elliptic curve, expressed as rational fractions, ensuring constant computation time and high efficiency, applicable in various cryptographic applications.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If probabilistic algorithms are used to insert arbitrary values using elliptic curves, then the implementation is simple and memory-efficient, but the implementation time is not constant and reveals information to attackers
Solution Approach 1:
The patent changes the fundamental parameter of the algorithm from probabilistic to deterministic by using the Euler function φ(A) where φ(A) mod 3 = 1. This parameter change enables constant-time computation while maintaining the elliptic curve-based cryptographic functionality, thereby resolving the contradiction between implementation simplicity and security against timing attacks
2Reliability
If useless steps are added to mask the time used by probabilistic insertion algorithm, then constant implementation time is achieved, but the processing becomes unwieldy and consumes a great deal of time
Solution Approach 1:
The patent extracts the essential functionality of constant-time computation from the Euler function φ(A) and integrates it directly into the cryptographic algorithm. This eliminates the need for separate useless masking steps, achieving both constant implementation time and computational efficiency without the overhead of additional dummy operations
3Loss of time
If deterministic functions expressed as rational fractions are used to obtain points on elliptic curve, then constant computation time is achieved, but the function must attain at least q/4I points
Solution Approach 1:
The patent demonstrates that the deterministic function based on the Euler function φ(A) with φ(A) mod 3 = 1 serves multiple purposes: it provides constant-time computation, achieves at least q/4I points, and maintains cryptographic security. This multi-functionality resolves the contradiction by showing that the same mathematical structure simultaneously satisfies all requirements
Data Source
AI summary
The method comprises, in an electronic component, carrying out a cryptographic calculation that includes the step of obtaining points P on an elliptic curve following the equation Y2+a1XY+a3Y=X3+a2X2+a4+X+a6 (1) where a1, a2, a3, a4 et a6 are elements of a set A of elements; where A is a ring of modular integers Z/qZ where q is a positive integer resulting from a number I of different prime numbers strictly higher than 3, I being an integer higher than or equal to 2, where A is a finite body Fq with q the power of a prime integer; where X and Y are the coordinates of the points P and are elements of A. The method comprises determining a diameter (11), and obtaining the coordinates X and Y of a point P (13) by applying a function (12) to said parameter. The Euler function φ of A corresponds to the equation φ(A) mod 3=1. The function is a reversible and deterministic function expressed by a rational fraction in a1, a2, a3, a4 and a6 and in said parameter in A, and reaches at least a number q/41 of points P, with I being equal to 1 for a finite body Pq. The method further comprises using the point P in a cryptographic application for ciphering or hashing or signature or authentication or identification.