Encrypted Multivariate Function Evaluation via Univariate Decomposition
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Solution Overview
Problem
Existing homomorphic encryption schemes are impractical for evaluating multivariate real-valued functions due to high computational complexity and the need for frequent bootstrapping operations, which are resource-intensive and slow, especially when dealing with functions having a complex Boolean circuit representation.
Innovation Solution
Transform multivariate functions into networks of univariate functions through sums and compositions, optimizing the evaluation process by identifying and reusing intermediate values, and representing functions as tables rather than Boolean circuits, thereby reducing computational complexity and time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If multivariate functions are evaluated using traditional Boolean circuit representation in homomorphic encryption, then the evaluation can be performed, but the computational complexity and computation time become prohibitively high
Solution Approach 1:
The patent segments multivariate functions into multiple univariate functions through variable substitution and decomposition. Each univariate function is then evaluated separately using homomorphic encryption, avoiding the need to evaluate the entire multivariate function as a single complex Boolean circuit. This segmentation reduces both computational complexity and evaluation time.
Solution Approach 2:
The patent introduces intermediate variables as mediators to transform the evaluation of multivariate functions into a sequence of univariate function evaluations. These intermediate variables serve as bridges between the original multivariate function and the simplified univariate evaluations, enabling efficient homomorphic computation while preserving the functional relationship.
2Reliability
If bootstrapping operations are performed frequently to maintain ciphertext noise levels, then the security and correctness of homomorphic encryption is preserved, but the computation time and resource consumption increase significantly
Solution Approach 1:
The patent performs preliminary decomposition of multivariate functions into univariate functions before homomorphic encryption. This preliminary action structure the computation in a way that reduces the number of subsequent bootstrapping operations needed, as univariate function evaluations generate less noise accumulation compared to evaluating complex multivariate Boolean circuits.
3Productivity
If multivariate functions are represented as Boolean circuits for homomorphic evaluation, then the evaluation can be performed, but the resource consumption and computation time become impractical
Solution Approach 1:
The patent segments the computational task by decomposing multivariate functions into univariate components. This segmentation allows each component to be evaluated with minimal computational resources using homomorphic encryption, avoiding the exponential resource consumption associated with evaluating full multivariate Boolean circuits.
Solution Approach 2:
The patent substitutes the mechanical Boolean circuit evaluation mechanism with a mathematical approach based on univariate function decomposition and homomorphic encryption. This substitution replaces the resource-intensive Boolean circuit evaluation with a more efficient mathematical framework that requires significantly fewer computational resources.
Data Source
AI summary
The invention relates to a cryptographic method and variants thereof based on homomorphic encryption enabling the evaluation of real-valued functions on encrypted data, in order to allow carrying out homomorphic processing on encrypted data more broadly and efficiently.