Given a primitive element of a finite field, the Zech logarithm relative to the base is defined by the equation
which is often rewritten as
The choice of base is usually dropped from the notation when it is clear from the context.
To be more precise, is a function on the integersmodulo 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 , that is for a field of characteristic 2, and for a field of odd characteristic with elements.
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 of 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.
For fields of characteristic 2,
.
Usesedit
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.
Examplesedit
Let α ∈ GF(23) be a root of the primitive polynomialx3 + x2 + 1. The traditional representation of elements of this field is as polynomials in α of degree 2 or less.
A table of Zech logarithms for this field are Z(−∞) = 0, Z(0) = −∞, Z(1) = 5, Z(2) = 3, Z(3) = 2, Z(4) = 6, Z(5) = 1, and Z(6) = 4. The multiplicative order of α is 7, so the exponential representation works with integers modulo 7.
Since α is a root of x3 + x2 + 1 then that means α3 + α2 + 1 = 0, or if we recall that since all coefficients are in GF(2), subtraction is the same as addition, we obtain α3 = α2 + 1.
The conversion from exponential to polynomial representations is given by
(as shown above)
Using Zech logarithms to compute α 6 + α 3:
or, more efficiently,
and verifying it in the polynomial representation:
^Zech, Julius August Christoph (1849). Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen (in German) (Specially reprinted (from Vega–Hülße collection) 1st ed.). Leipzig: Weidmann'sche Buchhandlung. Archived from the original on 2018-07-14. Retrieved 2018-07-14. Also part of: Freiherr von Vega, Georg (1849). Hülße, Julius Ambrosius [in German]; Zech, Julius August Christoph (eds.). Sammlung mathematischer Tafeln (in German) (Completely reworked ed.). Leipzig: Weidmann'sche Buchhandlung. Bibcode:1849smt..book.....V. Archived from the original on 2018-07-14. Retrieved 2018-07-14.
^Zech, Julius August Christoph (1863) [1849]. Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen (in German) (Specially reprinted (from Vega–Hülße collection) 2nd ed.). Berlin: Weidmann'sche Buchhandlung. Archived from the original on 2018-07-14. Retrieved 2018-07-13.
^Zech, Julius August Christoph (1892) [1849]. Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen (in German) (Specially reprinted (from Vega–Hülße collection) 3rd ed.). Berlin: Weidmann'sche Buchhandlung. Archived from the original on 2018-07-14. Retrieved 2018-07-13.
^Zech, Julius August Christoph (1910) [1849]. Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen (in German) (Specially reprinted (from Vega–Hülße collection) 4th ed.). Berlin: Weidmann'sche Buchhandlung. Archived from the original on 2018-07-14. Retrieved 2018-07-13.
Conway, John Horton (1968). Churchhouse, Robert F.; Herz, J.-C. (eds.). "A tabulation of some information concerning finite fields". Computers in Mathematical Research. Amsterdam: North-Holland Publishing Company: 37–50. MR 0237467.
Kühn, Klaus (2008). "C. F. Gauß und die Logarithmen" (PDF) (in German). Alling-Biburg, Germany. Archived (PDF) from the original on 2018-07-14. Retrieved 2018-07-14.