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Wikipedia / Zech's logarithm

Zech logarithms are used to implement addition in finite fields.

Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after Carl G. J. Jacobi who used them for number theoretic investigations.

Definition

Given a primitive element is usually dropped from the notation when it is clear from the context.

is a function on the integers modulo for a field of characteristic for a field of odd elements.

as a special case.

This can be extended to arithmetic of the projective line and other rules as appropriate.

Uses

For sufficiently small finite fields, a table of Zech logarithms allows an especially efficient implementation of all finite field arithmetic in terms of a small number of integer addition/subtractions and table look-ups.

The utility of this method diminishes for large fields where one cannot efficiently store the table. This method is also inefficient when doing very few operations in the finite field, because one spends more time computing the table than one does in actual calculation.

Examples

be a root of the primitive polynomial. The traditional representation of elements of this field is as polynomials of degree or less.

Here is a table of Zech logarithms for this field.

.

.

Visualization

Given a prime power and a primitive element:

  • at the center.

  • at the top.

  • .

.

are placed in counterclockwise order:

around the center node.

:

forms an Elementary abelian group (isomorphic to the additive group of the field). The rotational symmetry elegantly captures the algebraic indistinguishability of the non-zero elements in an elementary abelian group.

are roots of the same primitive polynomial is Euler's totient function distinct Zech logarithm tables for a given field size.

are reflections of each other.

).

:

See also

References

Further reading