Zech's logarithm

  (Redirected from Jacobi logarithm)

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.[6]

DefinitionEdit

Given a primitive element   of a finite field, the Zech logarithm relative to the base   is defined by the equation

 

or equivalently by

 

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 integers modulo 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.

Notice that for fields of characteristic two,

  .

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 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

  (as shown above)
 
 
 

Using Zech logarithms to compute α6 + α3:

 ,

or, more efficiently,

 ,

and verifying it in the polynomial representation:

 .

See alsoEdit

ReferencesEdit

  1. ^ 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; 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.
  2. ^ 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.
  3. ^ 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.
  4. ^ 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.
  5. ^ Lidl, Rudolf; Niederreiter, Harald (1997). Finite Fields (2nd ed.). Cambridge University Press. ISBN 978-0-521-39231-0.
  6. ^ Jacoby, Carl Gustav Jacob (1846). "Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie". Journal für die reine und angewandte Mathematik (in German). 1846 (30): 166–182. doi:10.1515/crll.1846.30.166. ISSN 0075-4102. (NB. Also part of "Gesammelte Werke", Volume 6, pages 254–274.)

Further readingEdit