Cryptographic Key Pair Integrity Verification via Candidate Factor Analysis
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Solution Overview
Problem
Current integrity tests for cryptographic key pairs may not detect errors effectively, particularly during key generation, which can lead to vulnerabilities in cryptographic systems and potential attacks.
Innovation Solution
A method for verifying the integrity of public and private key pairs by computing and verifying candidate factors p′ and q′, checking the congruence of the product of the public and private exponents modulo the least common multiple, and using probabilistic factorization algorithms to detect errors in key generation, while also simulating perturbations to test device resilience.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If current integrity tests (FIPS 140-2) are used for key pair verification, then the testing process is simple and fast, but the tests may not detect certain errors in key generation
Solution Approach 1:
The verification process is segmented into multiple independent checks: (1) checking if e*d ≡ 1 (mod λ(n)), (2) verifying the factorization n = p'*q', (3) checking if λ(n) is divisible by (p'-1) and (q'-1). Each segment targets specific error types, collectively providing comprehensive error detection while maintaining modular implementation that limits complexity growth.
Solution Approach 2:
The method performs preliminary verification of candidate factors p' and q' before final key acceptance. By computing λ'(n) from candidate factors and verifying e*d ≡ 1 (mod λ'(n)) in advance, the system detects generation errors before the keys are deployed, preventing vulnerable keys from entering service.
2Reliability
If additional verification steps are added to improve error detection, then reliability improves, but computational cost increases
Solution Approach 1:
The verification uses partial factorization information (candidate factors p' and q') rather than requiring complete prime factorization. By computing λ'(n) partially from candidate factors and performing modular verification, the method achieves substantial error detection with less computational effort than full factorization would require.
Solution Approach 2:
The method changes the verification parameter from checking only the cryptographic functionality (encryption/decryption) to also checking the mathematical structure (factorization relationship and λ(n) properties). This parameter expansion enables detection of additional error types without proportionally increasing computational cost, since the same candidate factors serve dual purposes.
3Object-affected harmful factors
If comprehensive verification of key pair integrity is performed, then security against attacks is improved, but the verification process becomes more complex
Solution Approach 1:
The verification process incorporates feedback loops where candidate factors p' and q' are tested, and if they fail verification (e*d ≠ 1 mod λ'(n)), the system rejects them and requests new candidates. This feedback mechanism ensures that only verified, secure key pairs are accepted, systematically eliminating vulnerable configurations without requiring complex preventive measures.
Solution Approach 2:
The candidate factors p' and q' serve as intermediaries between the key generation process and the final verification. By introducing these intermediate values that can be independently verified against the public key components, the system creates a verification bridge that simplifies the overall process compared to directly verifying the cryptographic properties of the full key pair.
Data Source
AI summary
Method of integrity verification of public and private cryptographic key pairs in the additive group of integers modulo n, with n being the product of two prime numbers p and q, the method including the following steps:of computation (201), on the basis of the number n, of a public exponent e of the public key, and of a private exponent d of the private key, of two candidate factors p′ and q′ corresponding respectively to the numbers p and q,of verification (206) so as to verify the consistency of the private exponent with respect to the public exponent and to the number n, the verification step involving the candidate factors.