Hermitian-Curve AMD Codes for Low-Complexity Manipulation Detection
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Solution Overview
Problem
Existing block cipher cryptographic algorithms are vulnerable to side-channel attacks, particularly in mobile devices, due to physical implementation flaws that leak information, and current error detection methods require large finite fields for effective protection against algebraic manipulation.
Innovation Solution
The use of algebraic curves to reduce the complexity and required field size of Algebraic Manipulation Detection (AMD) error detection codes, incorporating weak AMD codes and Hermitian curves to protect data before encryption, allowing for efficient protection of large data chunks with a low miss probability.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional AMD codes are used to protect data against algebraic manipulation, then detection capability is improved, but field size and computational complexity increase significantly
Solution Approach 1:
The patent changes the fundamental parameters of AMD codes by using small finite fields (such as GF(2)) instead of large finite fields. This is achieved by modifying the code construction to operate over small fields while maintaining strong detection capabilities through careful design of the encoding function and verification process. The parameter change from large to small field size directly resolves the contradiction between detection capability and computational complexity.
Solution Approach 2:
The patent segments the protection mechanism into multiple independent components: (1) encoding the message with redundancy over a small field, (2) using a specific algebraic structure for the encoding function, and (3) verifying integrity through field operations. This segmentation allows each component to be optimized independently, maintaining high detection capability while keeping the overall system complexity low through efficient small-field operations.
2Reliability
If redundancy is added to guarantee detection of worst-case errors, then detection reliability is improved, but data processing overhead increases
Solution Approach 1:
The patent changes the redundancy parameter by using compact redundancy representations over small finite fields. Instead of requiring large numbers of check bits, the invention achieves strong detection guarantees with fewer redundancy bits by utilizing the algebraic structure of small fields. This parameter change reduces the overhead ratio while maintaining high detection reliability for worst-case errors.
Solution Approach 2:
The patent replaces complex mechanical error detection mechanisms with algebraic operations over small finite fields. The verification process uses efficient field arithmetic operations (addition, multiplication, inversion) that can be implemented with simple logic circuits, substituting complex combinatorial checking mechanisms. This substitution maintains high detection reliability while significantly improving data processing throughput through faster, simpler operations.
Data Source
AI summary
A method for protecting encoded data from algebraic manipulation includes receiving a data word sεKd to be protected, randomly selecting two integers a ε{0, . . . , q−1} and b ε{0, . . . , √q−1}, finding a point (α, β) on a Hermitian curve over a field Fq that corresponds to the randomly selected integers (a, b) from a mapping (a, b)(α, β)=(ua, ua√q+1z+vb), whereua:={0ifa=0,γ1a-1otherwise,Vb:={0ifb=0,γ2b-1otherwise,and z is an element of the field Fq of unit trace, and where γ1 is a fixed primitive element of the field Fq and γ2 is a primitive element of a field F√q⊂Fq, and calculating a sum fs(α, β)=αi<sub2>d+1</sub2>βj<sub2>d+1</sub2>+Σk=1dαi<sub2>k</sub2>βkj<sub2>k </sub2>for a set of d+1 integers pairs I ={(ik,jk)}k=1d+1, where the encoded word is a triple (s, (α, β),fs(α, β)).