Optimal Extension Fields (OEFs)

Related Concepts

Finite Field; Mersenne Prime; Special Primes

Definition

Optimal extension fields (OEFs) are a family of finite fields with an arithmetic that can be implemented efficiently in software. OEFs are extension fields $\mathit{GF}({p}^{m})$ where the prime p is of special form.

Theory

OEFs were introduced first in [2] and independently in [7]. They are defined as follows:

Definition 1 An Optimal Extension Field is a finite field GF(p ^m) such that:

• 1. pis a prime number of the form ${2}^{n} \pm c{,\log }_{2}c \leq \left \lfloor \frac{1} {2}n\right \rfloor$ (such primes are also referred to as pseudo-Mersenne prime).

• 2. An irreducible binomial $P(x) = {x}^{m} - \omega$ exists over GF (p).

An example of an OEF is the field $\mathit{GF}({p}^{6})$ with the prime $p = {2}^{32} - 387$ and the irreducible polynomial ${x}^{6} - 2$. Note that the cardinality of this OEF is roughly ${({2}^{32} - 387)}^{6} \approx {2}^{192}$.

The main motivation for OEFs is that the...