Blind Rotation Bootstrapping Using NTT Polynomial Multiplication
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Solution Overview
Problem
Existing homomorphic encryption schemes, particularly those using programmable bootstrapping with blind rotation, are computationally expensive due to the need for polynomial multiplications modulo a modulus q=2Ω and quotient polynomial XN+1, limiting efficiency and scalability.
Innovation Solution
Implement polynomial multiplications using the number-theoretic transform (NTT) in a polynomial ring defined by a modulus q and a quotient polynomial p(X) that divides the NTT polynomial XM−1, allowing efficient computation of blind rotations and programmable bootstrapping.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If polynomial multiplications are performed modulo q=2^Ω and quotient polynomial X^N+1 in blind rotation, then encrypted computations can be performed, but computational overhead is high and efficiency is limited
Solution Approach 1:
The patent changes the parameter representation by introducing a transformed parameter β = α + 1, where α is the original ciphertext parameter. This parameter transformation enables the use of number-theoretic transform (NTT) for polynomial multiplication, converting the computational modulus from X^N+1 to a modulus suitable for NTT operations. The transformation maintains cryptographic security while enabling efficient FFT-based multiplication algorithms, thus resolving the contradiction between computational correctness and efficiency.
Solution Approach 2:
The patent replaces the traditional polynomial multiplication mechanism (direct convolution modulo X^N+1) with a number-theoretic transform-based mechanism. By substituting the mechanical polynomial multiplication process with NTT operations, the patent achieves O(N log N) complexity instead of O(N^2), dramatically improving computation efficiency while maintaining the correctness of encrypted computations through proper parameter transformation and modulus selection.
2Reliability
If programmable bootstrapping is implemented with traditional polynomial multiplication, then noise reduction is achieved, but computational complexity increases
Solution Approach 1:
The patent applies parameter transformation (β = α + 1) to enable NTT-based polynomial multiplication in the bootstrapping process. This transformation allows the system to perform the necessary polynomial operations with reduced complexity, making programmable bootstrapping more feasible. The parameter change maintains the noise reduction capability while significantly lowering the computational complexity barrier.
Solution Approach 2:
The patent substitutes traditional polynomial multiplication with NTT-based multiplication in the bootstrapping process. This replacement reduces the computational complexity from quadratic to near-linear time, making the noise reduction process more efficient and practical for real-world applications while preserving the fundamental bootstrapping functionality.
3Reliability
If blind rotation is performed with conventional polynomial multiplication methods, then ciphertext transformation is achieved, but scalability is limited
Solution Approach 1:
The patent replaces conventional polynomial multiplication with number-theoretic transform-based multiplication, enabling blind rotation operations to scale efficiently. The NTT-based approach reduces computational complexity from O(N^2) to O(N log N), allowing the system to handle larger ciphertexts and more complex operations, thus improving scalability while maintaining transformation correctness.
Solution Approach 2:
The patent introduces parameter transformation (β = α + 1) that enables the use of NTT in blind rotation operations. This parameter change allows the system to leverage efficient Fourier-based algorithms, improving scalability and adaptability for handling various ciphertext sizes and complexity requirements while preserving the correctness of the transformation process.
Data Source
AI summary
Some embodiments are directed to a cryptographic encrypted computation method (400). The method involves performing a blind rotation of a ciphertext according to a test polynomial. The blind rotation results in an encrypted polynomial product of the test polynomial and a bootstrapping monomial represents the plaintext value as an exponent, modulo a modulus (q) and modulo a quotient polynomial (p(X)). . . . The quotient polynomial p(X) divides a number-theoretic transform (NTT) polynomial XM−1 that allows a number-theoretic transform modulo the modulus q, e.g., q is a power of two and p(X)=XN+XN/2+1. The blind rotation is performed using the NTT, while the test polynomial is defined in such a way that the polynomial product is programmed to have desired output values for respective plaintext values as a fixed coefficient.