Homomorphic Evaluation of Real-Valued Functions with Univariate Decomposition
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Solution Overview
Problem
Current homomorphic encryption schemes are impractical for transforming a set of ciphertexts corresponding to cleartexts into a set of real-valued functions due to complex calculations and slow bootstrapping operations, particularly when dealing with multivariate functions, which are represented as Boolean circuits, leading to high computational complexity and time consumption.
Innovation Solution
Transform multivariate functions into networks of univariate functions through sums and compositions, optimizing the evaluation by identifying and reusing redundancies, and representing functions as tables rather than Boolean circuits, using homomorphic encryption schemes like LWE and RLWE to perform calculations on encrypted data.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If multivariate functions are represented as Boolean circuits for homomorphic evaluation, then the evaluation can be performed on encrypted data, but the computational complexity and time consumption increase significantly
Solution Approach 1:
The patent segments multivariate functions into multiple univariate functions through functional decomposition. Instead of evaluating a complex multivariate function f(x1, x2, ..., xn) directly as a Boolean circuit, the function is expressed as a composition of univariate functions g1, g2, ..., gm where each gi operates on a single variable. This segmentation reduces the multiplicative depth of the computation and allows for more efficient homomorphic evaluation while maintaining data confidentiality.
2Measurement precision
If bootstrapping operations are performed to reduce ciphertext noise, then the noise level decreases and calculation accuracy improves, but the computation time increases dramatically
Solution Approach 1:
The patent applies preliminary action by performing noise management and function decomposition before the main homomorphic evaluation process. The multivariate function is decomposed into univariate functions in advance, and noise thresholds are pre-calculated. This allows the system to perform fewer bootstrapping operations during the actual evaluation, as the decomposed structure inherently produces less noise accumulation compared to evaluating the original multivariate function directly.
3Measurement precision
If the number of bootstrapping operations is increased to maintain low noise levels, then calculation accuracy is maintained, but the productivity and processing speed decrease
Solution Approach 1:
By segmenting the multivariate function into univariate functions, the patent reduces the number of sequential operations required. Each univariate function evaluation produces less noise than the corresponding operation on the original multivariate function, thereby reducing the frequency of bootstrapping operations needed to maintain accuracy. This segmentation directly improves processing speed while maintaining noise control.
4Reliability
If multivariate functions are evaluated directly using existing homomorphic schemes, then the evaluation can be performed, but the storage and transmission capacities required increase
Solution Approach 1:
The patent segments the evaluation process into multiple simpler steps, each handling a single univariate function. This segmentation reduces the size of intermediate results and the computational state that must be stored and transmitted. Instead of maintaining the full state of a multivariate function evaluation, the system only needs to store and transmit results for individual univariate function evaluations, thereby reducing storage and transmission capacity requirements.
Data Source
AI summary
The invention relates to a cryptographic method and variants thereof based on homomorphic encryption enabling the evaluation of univariate or multivariate real-valued functions on encrypted data, in order to allow carrying out homomorphic processing on encrypted data more broadly and efficiently.