Galois Field Inversion via Sub-Field Segmentation
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Solution Overview
Problem
Conventional methods for computing multiplicative inverses in Galois fields, such as GF(28), require significant hardware resources and suffer from high latency, making them unsuitable for high-throughput systems like WLANs.
Innovation Solution
The method involves computing xpm+1, finding its inverse in GF(pm), and then multiplying by xpm to obtain the inverse of the element x in GF(p2m), utilizing a combination of linear and quadratic operations to reduce hardware complexity and latency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of time
If a look-up table is used to store multiplicative inverses for all 255 nonzero elements in GF(28), then latency is reduced, but hardware area and power consumption increase significantly
Solution Approach 1:
The patent segments the GF(28) inversion problem into operations in the sub-field GF(24). By representing elements of GF(28) as polynomials over GF(24) and using the relationship x^16+1 as the irreducible polynomial, the inversion is broken down into: (1) computing x^16+1 in GF(28), (2) finding the inverse of this element in the smaller field GF(24), and (3) combining results. This segmentation reduces the lookup table size from 255 entries to much smaller sub-field operations, significantly reducing hardware area while maintaining low latency.
2Area of stationary object
If the Extended Euclidean Algorithm is used to compute multiplicative inverses in GF(28), then hardware area is reduced, but latency increases
Solution Approach 1:
The patent changes the field parameter from GF(28) to its sub-field GF(24) for the inversion operation. By utilizing the mathematical property that if x is an element of GF(28), then x^16+1 is an element of the sub-field GF(24), the algorithm performs the computationally intensive inversion operation in the smaller field where latency is reduced, while still obtaining the correct inverse in the original GF(28) field through polynomial arithmetic.
3Reliability
If traditional techniques are used for Galois field inversion, then correctness is maintained, but hardware complexity increases
Solution Approach 1:
The patent applies the nested doll principle by embedding the GF(24) sub-field operations within the GF(28) field structure. Elements of GF(28) are represented as polynomials with coefficients in GF(24), and the inversion process nests the simpler GF(24) inversion algorithm inside the GF(28) computation framework. This nested approach maintains mathematical correctness for GF(28) while reducing hardware complexity by reusing the simpler sub-field operations.
Data Source
AI summary
A method and device for computing the multiplicative inverse of element x in Galois field GF(p2m) is proposed. In particular, when p is a prime number and m is an integer, the inverse may be constructed based on the observation that xp<sup2>m</sup2>+1 is en element in sub-field GF(pm) and the inverse of xp<sup2>m</sup2>+1 can be carried out in the sub-field. The inverse of X may be obtained by multiplying x−1=(xp<sup2>m</sup2>+1)−1 by xp<sup2>m</sup2>.