Zech's logarithm

Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator ${\displaystyle \alpha }$.

Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after C. G. J. Jacobi who used them for number theoretic investigations (C. G. J. Jacoby, "Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie", in Gesammelte Werke, Vol.6, pp. 254–274).

If ${\displaystyle \alpha }$ is a primitive element of a finite field, then the Zech logarithm relative to the base ${\displaystyle \alpha }$ is defined by the equation

or equivalently by

The choice of base ${\displaystyle \alpha }$ is usually dropped from the notation when it's clear from context.

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