Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator α {\displaystyle \alpha } . 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 α {\displaystyle \alpha } of a finite field, the Zech logarithm relative to the base α {\displaystyle \alpha } is defined by the equation
α Z α ( n ) = 1 + α n , {\displaystyle \alpha ^{Z_{\alpha }(n)}=1+\alpha ^{n},}
which is often rewritten as
Z α ( n ) = log α ( 1 + α n ) . {\displaystyle Z_{\alpha }(n)=\log _{\alpha }(1+\alpha ^{n}).}
The choice of base α {\displaystyle \alpha } is usually dropped from the notation when it is clear from the context. To be more precise, Z α {\displaystyle Z_{\alpha }} is a function on the integers modulo the multiplicative order of α {\displaystyle \alpha } , and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol − ∞ {\displaystyle -\infty } , along with the definitions
α − ∞ = 0 {\displaystyle \alpha ^{-\infty }=0}
n + ( − ∞ ) = − ∞ {\displaystyle n+(-\infty )=-\infty }
Z α ( − ∞ ) = 0 {\displaystyle Z_{\alpha }(-\infty )=0}
Z α ( e ) = − ∞ {\displaystyle Z_{\alpha }(e)=-\infty }
where e {\displaystyle e} is an integer satisfying α e = − 1 {\displaystyle \alpha ^{e}=-1} , that is e = 0 {\displaystyle e=0} for a field of characteristic 2, and e = q − 1 2 {\displaystyle e={\frac {q-1}{2}}} for a field of odd characteristic with q {\displaystyle q} elements. Using the Zech logarithm, finite field arithmetic can be done in the exponential representation:
α m + α n = α m ⋅ ( 1 + α n − m ) = α m ⋅ α Z ( n − m ) = α m + Z ( n − m ) {\displaystyle \alpha ^{m}+\alpha ^{n}=\alpha ^{m}\cdot (1+\alpha ^{n-m})=\alpha ^{m}\cdot \alpha ^{Z(n-m)}=\alpha ^{m+Z(n-m)}}
− α n = ( − 1 ) ⋅ α n = α e ⋅ α n = α e + n {\displaystyle -\alpha ^{n}=(-1)\cdot \alpha ^{n}=\alpha ^{e}\cdot \alpha ^{n}=\alpha ^{e+n}}
α m − α n = α m + ( − α n ) = α m + Z ( e + n − m ) {\displaystyle \alpha ^{m}-\alpha ^{n}=\alpha ^{m}+(-\alpha ^{n})=\alpha ^{m+Z(e+n-m)}}
α m ⋅ α n = α m + n {\displaystyle \alpha ^{m}\cdot \alpha ^{n}=\alpha ^{m+n}}
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