Hierarchical Group Key Management via Linear Geometry
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Solution Overview
Problem
Existing hierarchical group key management approaches face challenges in scalability, security, and efficiency, particularly in handling dynamic hierarchical structures and large numbers of subgroups, with existing methods either requiring significant computational resources or relying on complex cryptography.
Innovation Solution
A hierarchical group key management approach based on linear geometry in a vector space over a finite field, where a central controller assigns private vectors to subgroup controllers, maps them into confidential vectors, and uses public vectors to derive group keys, ensuring security and flexibility while minimizing computational and storage costs.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional cryptographic methods are used for hierarchical group key management, then security can be achieved, but computational and storage costs increase significantly
Solution Approach 1:
The patent replaces traditional cryptographic mechanisms (one-way functions, encryption algorithms) with a geometric algebraic system based on vector spaces and finite fields. Subgroup keys are derived through vector operations (inner products, linear combinations) rather than cryptographic computations, significantly reducing computational overhead while maintaining security through the mathematical properties of the geometric structure.
Solution Approach 2:
The patent changes the fundamental parameters of key management from cryptographic keys (large integers requiring complex operations) to geometric vectors (simpler algebraic structures). By representing subgroup keys as vectors in a finite field and using vector operations for key derivation, the system achieves the same security goals with much lower computational cost.
2Reliability
If complex cryptographic schemes are implemented, then security is improved, but device complexity increases
Solution Approach 1:
The patent substitutes complex cryptographic protocols with a unified geometric algebraic framework. Instead of implementing multiple cryptographic primitives (hash functions, encryption algorithms, digital signatures), the system uses a single consistent mathematical model based on vector spaces, where all key derivation and security operations follow the same geometric rules, simplifying the overall system architecture.
Solution Approach 2:
The geometric algebraic system serves multiple functions simultaneously: it provides key generation, key derivation, access control enforcement, and security verification all through vector operations. This multi-functional approach eliminates the need for separate cryptographic mechanisms for each function, reducing device complexity.
3Adaptability or versatility
If hierarchical structures are made dynamic to accommodate changing group compositions, then adaptability is improved, but key management complexity increases
Solution Approach 1:
The patent implements dynamic hierarchical structures through vector operations that naturally accommodate changes. When subgroups are added, removed, or reorganized, the system updates by performing vector additions, subtractions, or reconfigurations rather than complex cryptographic key updates. The geometric structure inherently supports dynamic reconfiguration while maintaining security relationships.
Solution Approach 2:
The patent changes the representation of hierarchical relationships from complex cryptographic dependency graphs to simple vector space relationships. By representing authorization relationships as vector linear dependencies and using basis vector transformations, the system achieves dynamic adaptability through simple parameter changes in the vector space rather than complex structural modifications.
Data Source
AI summary
A hierarchical group key management approach based on linear geometry is disclosed. The approach includes the following steps: step 1, the central controller selects a finite field F, a mapping parameter f and a constant N for use in the group; the central controller selects a N-dimensional private vector for each subgroup; step 2, the central controller selects a mapping parameter r and maps the private vector to a new set of vectors in the vector space; step 3, the central controller selects a subgroup key for each subgroup and constructs n linear systems of equations, and solves the solution of the linear equation systems, that is, the public vector, and the n sets of public vectors form a public vector; the public vector and the mapping parameter r are broadcasted or multicasted by the central controller to all the subgroup controllers; step 4, each subgroup controller solves the confidential vector of its own, and a set of key vectors is obtained by linear transformation of the confidential vector and the public matrix. This invention is simple and flexible, and is effective against brute-force attacks.