Homomorphic Ciphertext Comparison via Polynomial Approximation

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Solution Overview

Problem

Current homomorphic encryption methods only support basic operations like addition, multiplication, and lack the capability to perform non-polynomial computations such as comparison operations on encrypted messages, which can lead to resource wastage and security vulnerabilities.

Innovation Solution

A method and apparatus that utilize composite functions with complexity θ(α) or θ(α log α) to perform comparison computations like maximum, minimum, and size comparisons on homomorphic ciphertexts, using functions such as square root and sign functions, allowing for efficient computation without decrypting the data.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Reliability

If homomorphic encryption is used to perform computation on encrypted messages, then security is improved and decryption is avoided, but the capability to perform non-polynomial computation such as comparison computation is lost

Engineering Contradiction:

ImprovesecurityVSAvoidcomputation capability

Core Design Contradiction:

ReliabilityVSAdaptability or versatility

Solution Approach 1:

The patent transforms the comparison operation into an equivalent polynomial form by changing the mathematical parameters. Specifically, it uses the identity max(a,b) = (a+b+|a-b|)/2 and approximates the absolute value function |x| using a polynomial approximation (such as Taylor series or Chebyshev polynomials). This allows the comparison operation to be performed within the constraints of homomorphic encryption while maintaining security.

Inventive Principle:

Principle #35Parameter changes

Solution Approach 2:

The patent introduces intermediate computational steps that bridge the gap between non-polynomial comparison operations and the polynomial-only capabilities of homomorphic encryption. By breaking down the comparison into intermediate polynomial operations (addition, multiplication, and polynomial approximation of absolute value), the system achieves comparison functionality without directly supporting non-polynomial operations.

Inventive Principle:

Principle #24Intermediary (Mediator)

2Adaptability or versatility

If decryption is performed to enable comparison computation on messages, then computation capability is improved, but resource waste and security vulnerabilities increase

Engineering Contradiction:

Improvecomputation capabilityVSAvoidresource waste

Core Design Contradiction:

Adaptability or versatilityVSLoss of energy

Solution Approach 1:

The system enables comparison computation to be performed autonomously on encrypted data without requiring decryption. The homomorphic encryption scheme itself provides the computational capability needed for comparison through polynomial approximations, eliminating the need to switch to decryption mode and thereby avoiding the associated resource overhead and security risks.

Inventive Principle:

Principle #25Self-service

3Adaptability or versatility

If decryption is performed to enable comparison computation on messages, then computation capability is improved, but security vulnerabilities increase due to hacking risks

Engineering Contradiction:

Improvecomputation capabilityVSAvoidsecurity vulnerability

Core Design Contradiction:

Adaptability or versatilityVSObject-affected harmful factors

Solution Approach 1:

The patent applies homomorphic encryption beforehand to protect the data before any computation occurs. By enabling comparison operations directly on the encrypted data through polynomial approximations, the system cushions against potential security vulnerabilities that would arise from decryption, ensuring that sensitive data never exists in plaintext form during computation.

Inventive Principle:

Principle #11Beforehand cushioning (Prior cushioning)

4Adaptability or versatility

If polynomial approximation is used to enable comparison computation on homomorphic ciphertext, then adaptability is improved, but computational complexity increases

Engineering Contradiction:

Improvecomparison computation capabilityVSAvoidcomputational complexity

Core Design Contradiction:

Adaptability or versatilityVSDevice complexity

Solution Approach 1:

The patent uses polynomial approximation of the absolute value function with a finite number of terms rather than the exact non-polynomial form. By using a truncated polynomial series (such as 3-5 terms depending on required precision), the system achieves sufficient computational accuracy for comparison operations while keeping the computational complexity manageable and avoiding the need for infinitely complex exact representations.

Inventive Principle:

Principle #16Partial or excessive action

Data Source

PatentUS11799628B2Apparatus and method for processing non-polynomial operation on encrypted messages

Publication Date: 2023.10.24 CRYPTO LAB INC

AI summary

Disclosed is a ciphertext calculation method. The ciphertext calculation method comprises the steps of: receiving a comparative calculation command for a plurality of ciphertexts of the same type; performing a calculation by reflecting the plurality of ciphertexts of the same type on a synthesis function corresponding to the comparative calculation command; and outputting the calculated ciphertexts of the same type.