FHE Polynomial Evaluation Using Root Products and Ciphertext Rotations
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Solution Overview
Problem
Existing fully homomorphic encryption (FHE) schemes struggle to efficiently perform polynomial evaluations that require operations beyond addition, subtraction, and multiplication, such as division, square roots, and trigonometric functions, due to limitations in supporting non-scalar products and rotations that are not optimized.
Innovation Solution
A method for polynomial evaluation under FHE that involves generating and manipulating ciphertexts by duplicating input values, performing rotations, and scalar multiplications to enable operations not typically supported by FHE, specifically using the Cheon-Kim-Kim-Song scheme to support complex and real number operations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If existing FHE schemes use standard arithmetic operations (addition, subtraction, multiplication), then computations can be performed on encrypted data, but operations beyond these (division, square roots, trigonometric functions) cannot be efficiently performed
Solution Approach 1:
The patent changes the parameter space by introducing rotation operations and non-scalar products as new fundamental operations alongside standard arithmetic. This allows the FHE system to compute functions like division, square roots, and trigonometric functions by transforming them into polynomial evaluations using these extended operations, thereby achieving greater mathematical versatility without requiring entirely new cryptographic primitives for each function type.
Solution Approach 2:
The patent creates a universal evaluation framework where a single polynomial evaluation algorithm using rotations and non-scalar products can compute multiple different mathematical functions (division, square roots, trigonometric functions, activation functions) by appropriately choosing the polynomial and its coefficients. This multi-functional approach eliminates the need for separate specialized algorithms for each mathematical operation.
2Productivity
If polynomial evaluation uses standard FHE operations, then computations can be outsourced to untrusted parties, but the number of operations needed is not optimized
Solution Approach 1:
The patent segments the polynomial evaluation process into two distinct phases: (1) pre-computation of rotation matrices and non-scalar products offline, and (2) efficient evaluation online using these pre-computed components. This segmentation allows the computationally intensive parts to be prepared in advance, significantly reducing the number of operations required during actual polynomial evaluation and enabling faster computation even when outsourcing to untrusted parties.
Solution Approach 2:
The patent performs preliminary actions by pre-computing rotation matrices and non-scalar products before the actual polynomial evaluation. These pre-computed components are then used during evaluation to avoid repeating complex calculations, thereby optimizing the number of operations needed during the critical evaluation phase and improving overall productivity.
3Adaptability or versatility
If FHE supports rotations and non-scalar products, then operations like division and square roots become possible, but the cryptographic error accumulates
Solution Approach 1:
The patent uses the technique of copying and duplicating ciphertexts during the polynomial evaluation process. By working with multiple copies of the encrypted input value and systematically combining them through rotations and non-scalar products, the algorithm can compute higher-order terms and complex functions while maintaining control over error propagation. The duplicate copies allow for parallel computation paths that can be combined to achieve the desired mathematical operations with controlled error accumulation.
Data Source
AI summary
Evaluating polynomials for use under fully homomorphic encryption (FHE) is provided. An input polynomial of degree n is received, wherein n is equal to 2{circumflex over ( )}m. An input ciphertext containing an input value is also received. The input value is duplicated in n/2 slots. Two plaintext vectors each containing half of the roots in the polynomial are subtracted from the input ciphertext, obtaining second and third ciphertexts, which are multiplied elementwise to produce a result ciphertext comprising n/2 slots. The result ciphertext is rotated by 2{circumflex over ( )}i to generate a rotated ciphertext (i=iteration number) and multiplied by the rotated ciphertext to produce a new result ciphertext, for m−1 iterations. The final result ciphertext is multiplied with a leading coefficient of the polynomial, resulting in a final polynomial evaluation. An operation not supported under FHE is estimated according to the final evaluation.