NTT Polynomial Evaluation for Fault Detection in Cryptographic Operations
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Solution Overview
Problem
The security of RSA and elliptic-curve cryptographic systems is at risk due to the potential efficiency of quantum algorithms like Shor's algorithm, and there is a need for efficient methods to detect errors in number theoretic transforms used in lattice-based cryptography, particularly in post-quantum cryptographic algorithms such as Crystals-Kyber and Crystals-Dilithium.
Innovation Solution
A method is developed to validate the security of operations involving number theoretic transforms by comparing the results of polynomial evaluations before and after the transform, using techniques like polynomial interpolation to detect errors and ensure the integrity of computations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If polynomial evaluation and comparison methods are used to validate NTT, then fault detection capability is improved, but computational overhead increases
Solution Approach 1:
The patent applies partial verification by evaluating polynomials at specific points rather than verifying all coefficients. The method computes NTT of a polynomial and validates by checking equality at selected evaluation points, performing fewer operations than full verification while maintaining security. This partial action provides sufficient fault detection without the computational cost of complete validation.
Solution Approach 2:
The patent changes the verification parameter from checking all polynomial coefficients to checking at specific evaluation points. By transforming the verification problem from coefficient-space to evaluation-point-space, the method reduces computational complexity while maintaining the ability to detect faults in NTT operations.
2Reliability
If multiple evaluation points are used for validation, then security against quantum attacks is improved, but computational time increases
Solution Approach 1:
The patent uses a limited number of strategically selected evaluation points rather than exhaustive verification. By choosing specific points in the field where polynomial evaluation reveals NTT computation faults, the method achieves security validation with minimal computational time, balancing security requirements against performance constraints.
Data Source
Figure 1~2
AI summary
In the method for validating (VAL) the security of an operation involving a number theoretic transform (NTT) the number theoretic transform (NTT) is validated (VAL) carrying out the steps of - considering a given point of a field and evaluating a first value (f(c)) of a first polynomial (f) for the given point - Computing (CAL) a number theoretic transform of the polynomial - Computing (COMP) a second value of an interpolated (INT), second, polynomial of the number theoretic transform of the first polynomial for the given point and - Comparing (CHK) the first value (f(c)) and the second value (fint(c)) and - Concluding (VAL) that the number theoretic transform is valid if the first value (f(c)) and the second value (fint(c)) match and/or concluding (VAL) that the number theoretic transform is not valid if the first value (f(c)) and the second value (fint(c)) do not match.