Radix Decomposition for Homomorphic Encryption Noise Reduction
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Solution Overview
Problem
Homomorphic encryption schemes face challenges in managing noise levels during scalar multiplication, which limits the number of operations that can be performed before noise becomes too large, necessitating bootstrapping or reducing the number of operations.
Innovation Solution
Improving the radix decomposition method for scalar multiplication by allowing both plus and minus half the radix as digits, ensuring no two subsequent digits are equal in absolute value to half the radix, resulting in reduced noise and more efficient homomorphic computations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If standard radix decomposition is used for scalar multiplication, then the multiplication can be performed, but noise increases significantly limiting the number of operations
Solution Approach 1:
The patent applies local quality by making the radix decomposition non-uniform: most digits are chosen from {-B/2, ..., B/2} while allowing at most one digit to be B/2. This localized variation in digit selection strategy reduces noise accumulation at critical positions while maintaining overall multiplication capability.
Solution Approach 2:
The patent changes the parameters of radix decomposition by modifying the digit selection range and constraints. Instead of using a fixed digit range for all positions, it dynamically adjusts digit selection based on position, allowing {-B/2, ..., B/2} for most positions but restricting B/2 to at most one occurrence. This parameter change optimizes the noise properties of the homomorphic multiplication.
2Productivity
If more operations are performed on encrypted data, then computational capability increases, but noise accumulates making decryption impossible
Solution Approach 1:
The patent performs preliminary action by pre-optimizing the radix decomposition coefficients before homomorphic multiplication. By carefully selecting digits from the constrained set {-B/2, ..., B/2} with at most one B/2, the noise accumulation is predicted and minimized in advance, extending the number of operations that can be performed before bootstrapping is required.
3Productivity
If modulus parameters are reduced for efficiency, then computational performance improves, but noise management becomes more difficult
Solution Approach 1:
The patent changes the parameters of the multiplication algorithm itself (radix decomposition digits) to compensate for reduced modulus parameters. By using the optimized digit selection strategy with at most one B/2 digit, the noise growth is slowed, allowing smaller modulus parameters to be used while maintaining adequate noise margins for successful decryption.
Data Source
AI summary
Some embodiments are directed to a computer-implemented method (500) of determining a set of coefficients for homomorphically multiplying an encrypted value by a scalar. The encrypted value is represented by multiple respective value ciphertexts encrypting the value multiplied by respective powers of an even radix. The scalar multiplication is performed as a linear combination of the multiple respective value ciphertexts according to the set of coefficients. The set of coefficients are determined as digits of a radix decomposition of the scalar with respect to the radix. The determined digits lie between minus half the radix, inclusive, and plus half the radix, inclusive. It is ensured that no two subsequent digits are both equal in absolute value to half the radix.