Cryptographic Key Integrity Verification via Modular Congruence
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Solution Overview
Problem
Current cryptographic systems fail to reliably detect errors in the generation of pairs of public and private keys, which can lead to security breaches and attacks, as existing integrity tests do not effectively identify corrupted key pairs, especially during the calculation of the private exponent.
Innovation Solution
A method that verifies the coherence of the candidate private exponent with respect to the public exponent and the numbers p and q by performing additional modular multiplications, ensuring that the product of the candidate private exponent and the public exponent is congruent to specific values modulo (p-1) and (q-1), thereby enhancing the detection rate of key pair integrity issues.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If standard integrity tests (FIPS 140-2) are used to verify key pair generation, then the cryptographic system can detect some errors, but certain corrupted key pairs pass the tests undetected
Solution Approach 1:
The patent segments the verification process into two distinct phases: first verifying the relationship between private exponent d and public exponent e modulo λ(N), and second verifying the relationship modulo (λ(N)/α). This segmentation allows the system to detect errors that would pass through a single undivided verification step, thereby improving reliability without significantly increasing overall complexity.
Solution Approach 2:
The patent performs preliminary verification of the key pair relationship before the corrupted keys can be used for cryptographic operations. By checking the congruence relationships in advance, the system prevents undetected corrupted key pairs from being deployed, improving reliability while maintaining a manageable verification complexity through efficient modular arithmetic operations.
2Reliability
If the integrity test is repeated multiple times to achieve high detection rate (99% requires 14 repetitions), then the detection rate improves, but the processing cost increases significantly (at least 21,532 modular multiplications)
Solution Approach 1:
The patent performs a single preliminary verification using the relationship d·e ≡ 1 (mod λ(N)/α) that provides high detection capability without requiring multiple repetitions. This preliminary check with optimized modular arithmetic achieves near 100% detection rate while avoiding the cumulative processing cost of repeating standard tests 14 or more times, thus maintaining high productivity.
Solution Approach 2:
The patent changes the verification parameter from the standard single modulo λ(N) check to a two-stage verification involving modulo λ(N) and modulo (λ(N)/α). This parameter change enables the system to achieve superior detection rates with fewer operations, improving both reliability and productivity by avoiding the need for multiple repeated tests.
3Object-affected harmful factors
If an attacker causes errors in the calculation of the private exponent d by interfering with the device, then the attacker can recover secret factors (p and q) of N, but the standard integrity tests may not detect these errors
Solution Approach 1:
The patent applies preliminary anti-action by verifying the key pair relationship against the corrupted calculation result before the attacker can exploit the error. By checking whether d·e ≡ 1 (mod λ(N)/α) holds for the generated keys, the system detects attacks that would otherwise go unnoticed, countering the harmful effect of the interference and preventing secret factor recovery.
Solution Approach 2:
The patent introduces an intermediary verification step using the modular relationship with (λ(N)/α) that acts as a mediator between key generation and key usage. This intermediary check detects corrupted keys produced by attacks without requiring direct observation of the attack vector, thereby improving detection reliability while maintaining security against sophisticated interference attempts.
Data Source
AI summary
Disclosed is a method of verifying integrity of a pair of public and private cryptographic keys within the additive group of the integers modulo N, with N being the product of two primary numbers p and q, the method including: calculating a candidate private exponent d′ corresponding to a private exponent d of the private key; and executing a test of integrity. The test of integrity includes a step for verifying the coherence of the candidate private exponent d′ with respect to a public exponent e of the public key and to the numbers p and q, the verification step involving a first multiple modulo of the public exponent e of the public key and a second multiple modulo of the public exponent e of the public key.