Prime Number Generation via Base Decomposition and Residue Testing
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Solution Overview
Problem
Cryptographic devices face challenges in generating prime numbers efficiently due to the resource-intensive nature of GCD calculations, which can also pose security risks, especially on devices with limited processing capabilities.
Innovation Solution
A method for generating prime numbers using co-primality tests without GCD calculations, involving binary bases and small prime numbers, where the candidate prime is decomposed and tested for divisibility, allowing for efficient co-primality testing and reduced processing requirements.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If GCD calculation is used for co-primality testing, then the testing can be performed, but the processing power consumption increases and security risks arise
Solution Approach 1:
The patent extracts the essential requirement of co-primality testing (checking if candidate prime is divisible by small primes) from the complex GCD calculation process. By decomposing the candidate prime in base B and computing a simplified residue that preserves divisibility information, the method removes the computationally expensive GCD operation while maintaining testing accuracy.
Solution Approach 2:
The patent transforms the co-primality testing problem by changing the parameter representation. Instead of working with the original large candidate prime number in GCD calculations, the method changes to using a residue computed from base-B decomposition. This parameter transformation reduces the computational complexity from O(log n) GCD operations to simple modular arithmetic on smaller values.
2Reliability
If GCD calculation is used for co-primality testing, then the testing can be performed, but security risks increase due to potential leakage
Solution Approach 1:
The patent extracts only the necessary divisibility information from the candidate prime through base decomposition and residue computation. By taking out only the essential co-primality verification data (whether the residue is divisible by small primes) rather than performing full GCD calculations, the method eliminates security vulnerabilities associated with GCD leakage while preserving testing accuracy.
3Reliability
If rigorous probable primality test (Miller-Rabin) is used, then primality can be confirmed, but the processing cost becomes very high
Solution Approach 1:
The patent applies preliminary co-primality filtering using the efficient base decomposition method before performing the costly Miller-Rabin test. By pre-processing candidates through base-B decomposition and residue computation, the system eliminates composite numbers early in the process, so that only highly likely prime candidates proceed to the rigorous Miller-Rabin testing, thereby significantly improving overall processing efficiency.
Solution Approach 2:
The patent implements a two-stage testing approach where a lighter co-primality test (base decomposition + residue divisibility check) is applied to all candidates, and the full Miller-Rabin test is applied only when necessary. This partial application of testing intensity optimizes resource usage by performing excessive filtering at the first stage and reserved rigorous testing only for survivors.
Data Source
AI summary
The present invention relates to a method for generating a prime number and using it in a cryptographic application, comprising the steps of: a) determining at least one binary base B with a small size b=log2(B) bits and for each determined base B at least one small prime pi such that B mod pi=1, with i an integer, b) selecting a prime candidate YP, c) decomposing the selected prime candidate YP in a base B selected among said determined binary bases : YP=ΣyjBid) computing a residue yPB from the candidate YP for said selected base such that yPB=Σyje) testing if said computed residue yPB is divisible by one small prime pi selected among said determined small primes for said selected base B, f) while said computed residue yPB is not divisible by said selected small prime, iteratively repeating above step e) until tests performed at step e) prove that said computed residue yPB is not divisible by any of said determined small primes for said selected base B, g) when said computed residue yPB is not divisible by any of said determined small primes for said selected base B, iteratively repeating steps c) to f) for each base B among said determined binary bases, h) when, for all determined bases B, said residue yPB computed for a determined base is not divisible by any of said determined small primes for said determined base B, executing a known rigorous probable primality test on said candidate YP, and when the known rigorous probable primality test is a success, storing said prime candidate YP and using said stored prime candidate YP in said cryptographic application.