Non-Commutative Matrix Encryption Key Generation
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Solution Overview
Problem
Existing methods for generating common secret keys are not efficient in terms of speed and security, particularly in public key encryption systems that rely on finite commutative groups or non-commutative rings, and lack the ability to rapidly generate secure keys for high-speed data encryption and decryption.
Innovation Solution
A method involving the generation of non-commutative matrices A1 and A2 based on a common secret key, using public data and secret keys to compute matrices M A and M B, which are then used to derive a common secret key M AB, employing operations such as sum and product calculations with bit shifting and XOR operations to create pseudo-random number sequences for encryption and decryption.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If existing public key encryption systems employing finite commutative groups or non-commutative rings are used, then key exchange can be performed, but the generation speed of common secret keys is slow and cryptographic strength is insufficient
Solution Approach 1:
The patent changes the mathematical parameters from traditional finite commutative groups or simple non-commutative rings to a specific non-commutative ring structure defined by matrices over finite fields with determinant constraints. This parameter change enables both faster computation and enhanced cryptographic strength through the use of matrix operations and modular arithmetic with prime moduli
Solution Approach 2:
The patent combines multiple mathematical structures into a composite cryptographic system: it integrates non-commutative matrix rings, finite fields, modular arithmetic, and determinant-based security. This composite approach creates a cryptographic system that simultaneously achieves high-speed key generation through efficient matrix operations and strong security through the complexity of the combined mathematical structures
2Speed
If traditional public key encryption methods are employed, then secure communication can be established, but the encryption and decryption speed is limited
Solution Approach 1:
The patent replaces traditional mechanical-style modular exponentiation operations with matrix multiplication and inversion operations over finite fields. This substitution enables parallel computation and optimized linear algebra routines that execute faster while maintaining or enhancing security through the non-commutative matrix structure and determinant-based key derivation
3Adaptability or versatility
If conventional key generation methods are used, then key exchange can be performed, but the ability to handle variable bit lengths of encryption data is limited
Solution Approach 1:
The patent introduces dynamic adaptability by allowing the matrix dimension parameter d to vary based on the bit length requirements of the encryption data. The system dynamically selects appropriate matrix sizes and field parameters to match the data being protected, enabling efficient handling of variable-length messages without regenerating the entire key infrastructure
Data Source
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AI summary
Public data including a prime number p, a natural number d, a matrix Q, and a matrix S are acquired by a public data acquisition section (22), and secret key including natural numbers nA, kA is generated by a shared secret key generation section (24). A matrix MA (MA = S-kAQnASkA) is calculated by a non-commutative matrix generation section (26) and transmitted to a communication party, and a matrix MB (MB = S-kBQnBSkB) is acquired from the communication party. A matrix MAB (MAB = S-kAMBnASkA) is computed as a common secret key by a shared secret key computation section (28). An encryption/decryption device is thereby capable of rapid generation of the secure common secret key.