Definition
If is a primitive element of a finite field, then the Zech logarithm relative to the base is defined by the equation
or equivalently by
The choice of base is usually dropped from the notation when it's clear from context.
To be more precise, is a function on the integers modulo the multiplicative order of, and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol, along with the definitions
where is an integer satisfying .
Using the Zech logarithm, finite field arithmetic can be done in the exponential representation:
These formulas remain true with our conventions with the symbol, with the caveat that subtraction by is undefined. In particular, the addition and subtraction formulas need to treat as a special case.
This can be extended to arithmetic of the projective line by introducing another symbol satisfying and other rules as appropriate.
Read more about this topic: Zech's Logarithms
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