Optimal extension fields (OEFs) are a family of finite fields with special properties. They were designed in a way that leads to efficient field arithmetic if implemented in software. OEFs were introduced first in [3] and independently in [7]. They are defined as follows:
Definition 1. An Optimal Extension Field is a finite field\(\hbox{\it GF}(p^m)\) such that:
- 1.
p is aprime numberof the form\(2^n \pm c, \log_{2}c \leq\lfloor\frac{1}{2}n\rfloor\) (such primes are also referred to aspseudo-Mersenne prime),
- 2.
An irreducible binomial\(P(x) = x^m - \omega\)exists over GF\((p)\)}.
An example of an OEF is the field 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 field parameters can be chosen such that they are a good match for the processor on which the field arithmetic is to be implemented. In particular, it is often an...
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Paar, C. (2005). Optimal Extension Fields (OEFs). In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_289
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