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

TL;DR

This paper introduces a matrix decomposition method to reduce space complexity in odd-type Gaussian normal basis multipliers for binary fields, improving efficiency with minimal delay trade-offs.

Contribution

It presents a novel space reduction technique for odd-type Gaussian normal basis multipliers, extending matrix decomposition methods beyond optimal bases.

Findings

Reduces XOR gate count in implementations.

Applicable to 187 binary fields GF(2^k) with odd-type bases.

Achieves space savings with slight delay increase.

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.