Approximate Homomorphic Ring Encryption With Two-Variable NTT
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Solution Overview
Problem
Existing encryption algorithms are vulnerable to quantum attacks and face computational inefficiencies in performing arithmetic operations on encrypted data, particularly with large values of n.
Innovation Solution
A system for approximate homomorphic ring encryption that encodes messages in a two-variable polynomial ring, utilizing a ring learning with errors (RLWE) approach, enabling efficient polynomial multiplication through two-variable number theoretic transforms.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional encryption algorithms are used, then data security is provided, but computational efficiency deteriorates when performing arithmetic operations on encrypted data
Solution Approach 1:
The patent transitions from traditional one-variable polynomial rings to two-variable polynomial rings, adding a dimensional aspect to the encryption space. This allows messages to be encoded as bivariate polynomials M(X,Y) where operations can be performed homomorphically with reduced computational complexity, particularly for large values of n.
Solution Approach 2:
The patent replaces traditional mechanical/computational encryption operations with number theoretic transforms (NTT) in the two-variable polynomial ring. This substitution enables efficient arithmetic operations on encrypted data through algebraic transformations rather than direct computational manipulation, significantly improving productivity while maintaining security.
2Reliability
If traditional polynomial rings are used for encryption, then security is maintained, but computational complexity increases for large values of n
Solution Approach 1:
By introducing a second variable Y to the polynomial ring structure, the patent creates a two-variable polynomial ring qL[X,Y]/[Xn/2+1, Yn/2-(Xn/8-X3n/8)]. This dimensional expansion allows for more efficient representation and manipulation of encrypted data, reducing computational complexity while preserving security properties.
Solution Approach 2:
The patent modifies the polynomial ring parameters by introducing specific relations between X and Y variables and using carefully chosen moduli. These parameter changes enable number theoretic transforms to be applied more efficiently, reducing the computational burden for large values of n while maintaining the cryptographic security requirements.
3Measurement precision
If exact homomorphic encryption is used, then precise operation results are obtained, but computational burden increases significantly
Solution Approach 1:
The patent implements approximate homomorphic encryption rather than exact homomorphic encryption. This partial action approach provides sufficient precision for many practical applications while significantly reducing computational burden. The approximation is controlled through the choice of parameters in the two-variable polynomial ring and the number theoretic transform implementation.
Data Source
AI summary
Embodiments of the present application provide a system, a device, and a method for approximate homomorphic ring encryption. A message z, which is an (n/4)×(n/2) matrix may be encoded to a polynomial M(X, Y) in a polynomial ring =[X, Y]/Xn/2+1, Yn/2−(Xn/8−X3n/8). The polynomial M(X, Y) may then be encrypted to a ciphertext and sent to a target device. Encoding in the polynomial ring allows at least approximate homomorphic manipulation of the subsequent ciphertext for improved security.