Lattice-Based Cryptography Polynomial Multiplication Efficiency
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Solution Overview
Problem
Current lattice-based cryptography faces challenges in achieving efficient performance and robustness against side-channel attacks, particularly in the implementation of cryptographic operations on hardware devices like microcontrollers and integrated circuits, where power consumption and physical behavior can be exploited by attackers.
Innovation Solution
The method involves sampling two polynomials, determining specific coefficients for each, and performing polynomial multiplication in a ring Rq=Zq[x]/(xn+1), using techniques like schoolbook, Karatsuba, or Number Theoretic Transform (NTT) multiplication to reduce the number of random numbers required, thereby enhancing efficiency and security against side-channel attacks.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If polynomial multiplication is performed in lattice-based cryptography to achieve cryptographic operations, then the security and functionality are improved, but the consumption of random numbers increases and the operation becomes more vulnerable to side-channel attacks
Solution Approach 1:
The patent extracts only the necessary k coefficients from the second polynomial while setting the remaining coefficients to zero. This selective extraction reduces the random number consumption from what would be required for a full polynomial to just what is needed for the cryptographic operation, directly resolving the contradiction between security requirements and random number usage.
Solution Approach 2:
Instead of using complete polynomials with all coefficients set, the patent applies partial action by determining only k coefficients of the second polynomial and setting the rest to zero. This partial determination achieves the cryptographic security requirement while significantly reducing the amount of random numbers needed, as fewer coefficients require random value assignment.
2Reliability
If polynomial multiplication is performed with full coefficients to ensure cryptographic security, then the security is improved, but the computational complexity and processing time increase
Solution Approach 1:
The patent extracts only the essential k coefficients from the second polynomial for the cryptographic operation. By removing unnecessary coefficients and setting them to zero, the computational complexity of polynomial multiplication is reduced while maintaining the security requirements of the lattice-based cryptographic scheme.
Solution Approach 2:
The patent applies partial action by determining only k coefficients rather than all coefficients of the polynomial. This partial determination reduces the number of multiplication operations and computational steps required, thereby reducing device complexity and processing time while still achieving the necessary cryptographic security through the structured use of these k coefficients.
3Productivity
If more coefficients are determined in the second polynomial to reduce random number usage, then the efficiency is improved, but the vulnerability to side-channel attacks may increase
Solution Approach 1:
The patent changes the parameter structure by determining exactly k coefficients of the second polynomial and setting the remaining coefficients to zero. This parameter change creates a structured sparse polynomial that improves efficiency by reducing operations while simultaneously mitigating side-channel attack vulnerability through the systematic zeroing of unnecessary coefficients, which creates a more predictable and less variable operational pattern.
Data Source
AI summary
A method for executing a cryptographic operation is provided comprising acts comprising: (i) sampling a first polynomial, wherein one or more (e.g., one, some and/or all) coefficients of the first polynomial are determined; (ii) sampling a second polynomial, wherein a selection of k coefficients of the second polynomial is determined; (iii) multiplying the first polynomial with the second polynomial to determine a result; and (iv) using the result of the multiplication in the cryptographic operation. A security device arranged to perform one, some and/or all of the acts is provided.