Finite Group Verification Apparatus Using Integer Partitioning
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Solution Overview
Problem
In cryptographic systems employing finite groups, the large order of these groups leads to prolonged processing times for group operations, making it inefficient to verify whether h = s·e or h = se.
Innovation Solution
A verification apparatus comprising a processing device, integer partitioning unit, verification value calculation unit, and verification judging unit, which calculates and judges the consistency of h = s·e by partitioning the integer e into e1 and e2 such that e1·e ≡ e2 (mod p), with e1 and e2 being less than the square root of the order p, thereby reducing the time required for verification.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of time
If group operation is performed directly to verify h = s·e, then verification accuracy is maintained, but processing time becomes excessively long due to large group order
Solution Approach 1:
The patent segments the verification process by introducing an intermediate value a = e1·h - e2·s, where e1 and e2 are derived from partitioning the exponent e. This segmentation transforms the direct verification into a two-step process: first computing the intermediate value with smaller coefficients, then checking if it equals the identity element. This reduces the computational complexity and time required for verification while maintaining accuracy.
Solution Approach 2:
The patent changes the parameters of the verification equation by introducing new integers e1 and e2 such that e1·e ≡ e2 (mod p), where |e1| and e2 are smaller than the original exponent e. This parameter transformation allows the verification to use smaller coefficients in the group operation, significantly reducing the processing time while preserving the verification result.
2Reliability
If the order p of the finite group is made very large for security, then cryptographic strength is improved, but group operation processing time increases significantly
Solution Approach 1:
The patent segments the exponent e into components e1 and e2 through the relation e1·e ≡ e2 (mod p), where |e1| and e2 are smaller than e. This segmentation allows the verification to perform group operations with smaller coefficients, reducing the time required for each operation while maintaining the security provided by the large order p.
Solution Approach 2:
The patent introduces an intermediary value a = e1·h - e2·s as a mediator in the verification process. This intermediate computation uses smaller coefficients and can be performed more quickly, serving as a bridge to the final verification step without requiring direct computation with the full exponent e, thus reducing overall processing time while preserving security.
Data Source
AI summary
An integer partitioning unit inputs an order p of a finite group G and an integer e, and calculates an integer e1 and an integer e2 that satisfy e1·e≡e2 (mod p) based on the order p of the finite group G and the integer e which are input. A verification value calculation unit inputs an element s of the finite group G and an element h of the finite group G, and calculates an element a (=e1·h−e2·s) of the finite group G based on the element s and the element h which are input and the integer e1 and the integer e2 which are calculated by the integer partitioning unit in the integer partitioning process. A verification judging unit judges, based on the element a calculated by the verification value calculation unit, whether or not the element a is an identity element O of the finite group G. Hence, whether or not h =e·s is judged at high speed.