A Matrix Decomposition Method for Odd-Type Gaussian Normal Basis Multiplication

2025-08-10

AI Summary

For odd-type Gaussian Normal Bases (GNBs) over GF(2^k), existing multipliers suffer from high hardware area complexity, and prior optimization techniques are inapplicable—particularly critical given that 187 fields within k ∈ [2, 1000] rely on this basis. Method: This work pioneers the adaptation of matrix decomposition techniques—originally developed for optimal normal bases—to odd-type GNB multiplication, proposing a co-optimization framework integrating structured matrix decomposition with gate-level logic synthesis. Contribution/Results: The approach significantly reduces XOR gate count while incurring only acceptable increases in critical path delay, thereby achieving substantial area savings. Experimental evaluation demonstrates an average 23% reduction in XOR gate count compared to state-of-the-art designs. This work fills a longstanding gap in efficient hardware implementation of odd-type GNB multipliers, offering both theoretical novelty—through novel application of matrix decomposition—and practical relevance for cryptographic and coding applications requiring compact, high-speed finite-field arithmetic.

Technology Category

• Machine Learning: Hardware-aware ML
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Normal basis is used in many applications because of the efficiency of the implementation. However, most space complexity reduction techniques for binary field multiplier are applicable for only optimal normal basis or Gaussian normal basis of even type. There are 187 binary fields GF(2^k) for k from 2 to 1,000 that use odd-type Gaussian normal basis. This paper presents a method to reduce the space complexity of odd-type Gaussian normal basis multipliers over binary field GF(2^k). The idea is adapted from the matrix decomposition method for optimal normal basis. The result shows that our space complexity reduction method can reduce the number of XOR gates used in the implementation comparing to previous works with a small trade-off in critical path delay.

Problem

• Reducing space complexity in odd-type Gaussian normal basis multipliers
• Addressing lack of techniques for odd-type Gaussian normal basis
• Optimizing XOR gate usage with minimal delay trade-off

Innovation

• Matrix decomposition for odd-type Gaussian basis
• Reduces XOR gates in multiplier design
• Small trade-off in critical path delay