SIMD Matrix Determinant Computation Under FHE Noise Constraints
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Solution Overview
Problem
Existing techniques for computing matrix determinants under fully homomorphic encryption (FHE) are inefficient due to the inherent growth of noise in ciphertexts and reliance on conditional operations that are difficult to perform under encryption, making them unsuitable for FHE environments.
Innovation Solution
Leveraging single instruction multiple data (SIMD) processing to compute determinants of sub-matrices in parallel, splitting the matrix into halves and using SIMD operations to efficiently compute determinants of each half, then combining the results to obtain the final determinant.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional determinant computation techniques are used under FHE, then computational correctness is maintained, but computational efficiency deteriorates due to noise growth and conditional operations
Solution Approach 1:
The matrix is divided into sub-matrices, and the determinant computation is segmented into multiple manageable parts. Each sub-matrix determinant is computed separately using FHE, and the results are combined through multiplication and addition operations. This segmentation reduces the overall computational complexity and noise growth compared to computing the full matrix determinant directly.
Solution Approach 2:
Multiple sub-matrix determinants are merged together through homomorphic multiplication and addition operations to obtain the final determinant result. The patent combines the results of individual sub-determinant computations using the mathematical relationship that the full matrix determinant equals the product of diagonal elements from the upper triangular matrix plus the product of diagonal elements from the lower triangular matrix.
2Ease of operation
If conditional operations are used in determinant computation, then algorithmic correctness is maintained, but ease of operation deteriorates due to difficulty of performing conditional operations under encryption
Solution Approach 1:
The patent extracts and eliminates conditional operations from the determinant computation algorithm. Instead of using conditional logic that would require decryption or complex encrypted comparisons, the method uses direct algebraic operations (multiplication and addition) on encrypted sub-matrix determinants, which are naturally supported by FHE schemes.
Solution Approach 2:
The patent replaces the mechanical/algorithmic conditional operation system with a mathematical algebraic system. The determinant computation is transformed from a process requiring conditional branching into a process using only homomorphic multiplication and addition, substituting complex control flow with simpler mathematical operations that are native to FHE.
3Measurement precision
If noise growth is allowed to accumulate during computation, then computational simplicity is maintained, but measurement precision deteriorates due to ciphertext becoming undecryptable
Solution Approach 1:
The computation is segmented into smaller sub-matrix determinant calculations rather than computing the full matrix determinant in one step. This segmentation limits the accumulation of noise in each individual computation step, keeping the noise level manageable and the ciphertext decryptable throughout the process.
Solution Approach 2:
The patent performs preliminary decomposition of the matrix into sub-matrices before the main determinant computation. By pre-processing the matrix into a form where smaller determinants can be computed, the noise growth is controlled from the outset, preventing it from reaching levels that would make decryption impossible.
Data Source
AI summary
A method, apparatus and computer program product for homomorphic computation enables secure computation of determinants of a matrix under Fully Homomorphic Encryption (FHE). According to this disclosure, encrypted data that contains the values of a matrix is received at a server. The matrix is separated into at least a first portion, and a second portion. Each portion is configured as a square. A first data vector of ciphertext is computed for the first portion, and a second data vector of ciphertext is computed for the second portion. Under FHE, determinants of the first and second data vectors are computed as Single Instruction Multiple Data (SIMD) operations to generate a set of results. The set of results are then used to compute a determinant of the matrix. The determinant may then be used for FHE-based analytics.