Zech's Logarithms - Examples

Examples

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.

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

\alpha^3 = \alpha^2 + 1 (as shown above)
\alpha^4 = \alpha^3 \alpha = (\alpha^2 + 1)\alpha = \alpha^3 + \alpha = \alpha^2 + \alpha + 1
\alpha^5 = \alpha^4 \alpha = (\alpha^2 + \alpha + 1)\alpha = \alpha^3 + \alpha^2 + \alpha = \alpha^2 + 1 + \alpha^2 + \alpha = \alpha + 1
\alpha^6 = \alpha^5 \alpha = (\alpha + 1)\alpha = \alpha^2 + \alpha

Using Zech logarithms to compute α6 + α3:

and verifying it in the polynomial representation:

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