Memory Address Scrambling Using Galois-Field Bijective Maps
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Solution Overview
Problem
Existing memory devices face security vulnerabilities due to the limited number of circulant reordering matrices available for address scrambling, making it easier for intercepting devices to determine logical to physical address mappings.
Innovation Solution
The use of bijective matrices, including those generated using Galois fields, to increase the variety of matrices available for address scrambling, thereby increasing the time required for an intercepting device to determine the correct mapping.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If circulant reordering matrices are used for address scrambling, then the address mapping can be implemented, but the security is weak due to the limited number of available matrices
Solution Approach 1:
The patent changes the mathematical field parameter from standard arithmetic to Galois field arithmetic. This parameter change enables the use of bijective matrices which have the property of being both one-to-one and onto, significantly increasing the number of available scrambling matrices while maintaining the security function.
Solution Approach 2:
The patent introduces Galois field arithmetic as an intermediary mathematical system between the address space and the scrambling operation. This intermediary provides a structured framework for performing matrix operations that guarantee bijectivity, thereby solving the limitation of circulant matrices while maintaining computational feasibility.
2Reliability
If more matrices are used for address scrambling, then security increases, but the device complexity increases
Solution Approach 1:
By changing to Galois field arithmetic, the patent achieves a more efficient mathematical structure where matrix multiplication and inversion have well-defined properties. This parameter change allows for a larger set of usable matrices without proportionally increasing hardware complexity, as the Galois field operations can be implemented efficiently.
Solution Approach 2:
The patent replaces the mechanical approach of using only circulant matrices with a mathematical field-based approach using Galois field arithmetic. This substitution allows for more matrices to be used without requiring proportionally more physical hardware, as the complexity is managed through mathematical structure rather than physical multiplication.
Data Source
AI summary
Methods, systems, and devices for address scrambling by linear maps in Galois fields are described. For instance, a device may determine a bijective matrix based on a power up condition. In some examples, the device may determine the bijective matrix based on a seed value and/or may select the matrix from among a set of bijective matrices. In some examples, the bijective matrix may have at least one column and/or one row that has at least two non-zero elements. The device may generate a first address of a first address space based on applying the matrix (e.g., each column of the matrix) to at least a portion of a second address of a second address space and may access a memory array of the device based on generating the first address.