Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator
Zech logarithms are named after Julius Zech,[1][2][3][4] and are also called Jacobi logarithms,[5] after Carl G. J. Jacobi who used them for number theoretic investigations.
of a finite field, the Zech logarithm relative to the base
In order to describe every element, it is convenient to formally add a new symbol
Using the Zech logarithm, finite field arithmetic can be done in the exponential representation: These formulas remain true with our conventions with the symbol
This can be extended to arithmetic of the projective line by introducing another symbol
For fields of characteristic 2, 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.
Let α ∈ GF(23) be a root of the primitive polynomial x3 + x2 + 1.
The traditional representation of elements of this field is as polynomials in α of degree 2 or less.
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 Using Zech logarithms to compute α 6 + α 3: or, more efficiently, and verifying it in the polynomial representation: