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Zech's logarithms are used with finite fields to reduce a high-degree polynomial that is not in the field to an element in the field (thus having a lower degree). Unlike the traditional logarithm, the Zech's logarithm of a polynomial provides an equivalence — it does not alter the value.## Examples

### Polynomial basis

Let α ∈ GF(2^{3}) be a root of the primitive polynomial x^{3} + x^{2} + 1. Thus all powers of α higher than 2 can be reduced.### Normal basis

The normal basis representation of elements in this set will only use the 3 elements β, β^{2}, and β^{4}. We can see by looking at the above example that if we set β = α then β^{2} = α^{2} and β^{4} = α^{2} + α + 1, and thus β, β^{2}, and β^{4} are linearly independent and form a normal basis. So all elements in the field can be written as linear combinations of β, β^{2}, and β^{4}.

Let $alpha$ be a primitive element of a finite field, then $Z(n)$, the Zech logarithm of an integer $n$ may be defined such that

- $$

That is, $$

Z(n) = log(1 + alpha^n)where the logarithm is taken to the base $alpha$. Note that if $alpha^n$ is the minus one element of the field, then $Z(n)$ is undefined (since that would involve the logarithm of zero).

Since α is a root of x^{3} + x^{2} + 1 then that means α^{3} + α^{2} + 1 = 0, or if we recall that since all coefficients are in GF(2), subtraction is the same as addition, we obtain α^{3} = α^{2} + 1.

Now we can easily reduce the set

- $$

by the primitive polynomial as such:

- $$

- $$

- $$

- $$

These polynomials are known as the Zech's logarithms for their corresponding powers of α. The representation of all elements of GF(2^{3}) is

- $$

We find that, using similar calculations to those above, that the Zech's logarithms for

- $$

are equal to

- $$

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Last updated on Sunday August 31, 2008 at 22:14:34 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Sunday August 31, 2008 at 22:14:34 PDT (GMT -0700)

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