Unique-Spiral Data Representation for Linear-Time FHE Operations
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Solution Overview
Problem
Existing technologies face significant computational inefficiencies in performing mathematical operations on encrypted data, particularly in Fully Homomorphic Encryption (FHE), Post-Quantum Cryptography (PQC), and Artificial Intelligence (AI), due to the inability to efficiently perform polynomial operations like multiplication and addition, which results in dramatically longer runtimes compared to unencrypted calculations.
Innovation Solution
The method transforms data into a unique-spiral representation, allowing operations like addition and multiplication to be performed in linear runtime (O(K)), eliminating the need for time-consuming transformations like NTT and INTT, thereby reducing runtime by converting data into polynomial coefficients and using a transformation matrix to switch between representations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional FHE techniques are used to perform operations on encrypted data, then data security is maintained, but computational efficiency deteriorates significantly with runtimes increasing from less than an hour to more than a year
Solution Approach 1:
The patent changes the representation parameters of encrypted data from traditional polynomial forms to a new algebraic structure that enables efficient operations. By transforming the mathematical representation while maintaining encryption properties, the system achieves both security and efficiency.
Solution Approach 2:
The patent replaces the traditional mechanical approach of encrypting data and then performing slow homomorphic operations with a substituted approach using a new algebraic structure that naturally supports efficient operations while maintaining security properties.
2Speed
If NTT transform is used for efficient polynomial multiplication in FHE, then multiplication speed is improved, but the need for continual NTT and INTT transforms for addition operations increases overall computational overhead
Solution Approach 1:
The patent extracts and eliminates the need for repeated NTT and INTT transforms by working directly in a new algebraic representation that supports both addition and multiplication efficiently without requiring transformation between different spaces.
Solution Approach 2:
The new algebraic structure serves multiple functions simultaneously - it enables both efficient addition and multiplication operations without requiring separate transformation steps, making the system more universal and efficient.
3Reliability
If FHE calculations are performed using existing hardware and techniques, then data protection during computation is achieved, but the calculation time increases from less than an hour to more than a year
Solution Approach 1:
The patent fundamentally changes the mathematical parameters and representation of encrypted data to enable fast operations. By using a new algebraic structure instead of traditional polynomial representations, the system maintains data protection while reducing calculation time dramatically.
4Measurement precision
If polynomial operations are performed in traditional representation for FHE, then mathematical correctness is maintained, but computational efficiency deteriorates due to inability to perform operations in linear runtime
Solution Approach 1:
The patent substitutes the traditional polynomial-based mechanical system with a new algebraic structure that maintains mathematical correctness while enabling linear runtime operations through its inherent properties.
Data Source
AI summary
Systems, devices, software, and methods of the present invention provide for homomorphically encrypted (HE) and other data represented as polynomials of degree K-1 to be transformed in 0(K*log (K)) time into ‘unique-spiral’ representations in which both linear-time (0(K)) addition and linear-time multiplication are supported without requiring an intervening transformation. This capability has never previously been available and enables very significant efficiency improvements, i.e., reduced runtimes, for applications such as Fully Homomorphic Encryption (FHE), Post-Quantum Cryptography (PQC) and Artificial Intelligence (AI). Other efficient operations, such as polynomial division, raising to a power, integration, differentiation and parameter-shifting are also possible using the unique-spiral representations. New methods are introduced based on the unique-spiral representation that have applications to efficient polynomial composition, inversion, and other important topics.