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| '''Zech logarithms''' are used to implement addition in [[finite field]]s when elements are represented as powers of a generator <math>\alpha</math>. | | Hello, my title is Felicidad but I don't like when individuals use my full name. Bookkeeping has been his working day job for a whilst. Arizona is her birth location and she will by no means transfer. What she enjoys performing is to play croquet but she hasn't made a dime with it.<br><br>my webpage :: extended car warranty ([http://berkshirewatergardens.com/UserProfile/tabid/42/userId/270007/Default.aspx visit my website]) |
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| Zech logarithms are named after Julius Zech, and are also called Jacobi Logarithm,<ref>{{Citation | last1=Lidl | first1=Rudolf | last2=Niederreiter | first2=Harald | title=Finite fields | publisher=[[Cambridge University Press]] | isbn=978-0-521-39231-0 | year=1997}}
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| </ref> after Jacobi who used them for number theoretic investigations (C. G. J. Jacoby, "Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie", in Gesammelte Werke, Vol.6, pp. 254–274).
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| ==Definition==
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| If <math>\alpha</math> is a [[Primitive element (finite field)|primitive element]] of a finite field, then the Zech logarithm relative to the base <math>\alpha</math> is defined by the equation
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| :<math>Z_\alpha(n) = \log_\alpha(1 + \alpha^n),</math>
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| or equivalently by
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| :<math>\alpha^{Z_\alpha(n)} = 1 + \alpha^n.</math>
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| The choice of base <math>\alpha</math> is usually dropped from the notation when it's clear from context.
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| To be more precise, <math>Z_\alpha</math> is a function on the integers modulo the multiplicative order of <math>\alpha</math>, and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol <math>-\infty</math>, along with the definitions
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| :<math>\alpha^{-\infty} = 0</math>
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| :<math>n + (-\infty) = -\infty</math>
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| :<math>Z_\alpha(-\infty) = 0</math>
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| :<math>Z_\alpha(e) = -\infty</math>
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| where <math>e</math> is an integer satisfying <math>\alpha^e = -1</math>.
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| Using the Zech logarithm, finite field arithmetic can be done in the exponential representation:
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| :<math>\alpha^m + \alpha^n = \alpha^m \cdot (1 + \alpha^{n-m}) = \alpha^m \cdot \alpha^{Z(n-m)} = \alpha^{m + Z(n-m)} </math>
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| :<math>-\alpha^n = (-1) \cdot \alpha^n = \alpha^e \cdot \alpha^n = \alpha^{e+n}</math>
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| :<math>\alpha^m - \alpha^n = \alpha^m + (-\alpha^n) = \alpha^{m + Z(e+n-m)} </math> | |
| :<math>\alpha^m \cdot \alpha^n = \alpha^{m+n}</math>
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| :<math>\left( \alpha^m \right)^{-1} = \alpha^{-m}</math> | |
| :<math>\alpha^m / \alpha^n = \alpha^m \cdot \left( \alpha^n \right)^{-1} = \alpha^{m - n}</math>
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| These formulas remain true with our conventions with the symbol <math>-\infty</math>, with the caveat that subtraction by <math>-\infty</math> is undefined. In particular, the addition and subtraction formulas need to treat <math>m = -\infty</math> as a special case.
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| This can be extended to arithmetic of the [[projective line]] by introducing another symbol <math>+\infty</math> satisfying <math>\alpha^{+\infty} = \infty</math> and other rules as appropriate.
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| ==Uses==
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| 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.
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| 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 you spend more time computing the table than you do in actual calculation.
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| ==Examples==
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| Let α ∈ GF(2<sup>3</sup>) be a root of the [[Primitive polynomial (field theory)|primitive polynomial]] ''x''<sup>3</sup> + ''x''<sup>2</sup> + 1. The traditional representation of elements of this field is as polynomials in α of degree 2 or less.
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| 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.
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| Since α is a root of ''x''<sup>3</sup> + ''x''<sup>2</sup> + 1 then that means α<sup>3</sup> + α<sup>2</sup> + 1 = 0, or if we recall that since all coefficients are in GF(2), subtraction is the same as addition, we obtain α<sup>3</sup> = α<sup>2</sup> + 1.
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| The conversion from exponential to polynomial representations is given by
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| :<math>
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| \alpha^3 = \alpha^2 + 1</math> (as shown above)
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| :<math>
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| \alpha^4 = \alpha^3 \alpha = (\alpha^2 + 1)\alpha = \alpha^3 + \alpha = \alpha^2 + \alpha + 1
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| </math>
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| :<math>
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| \alpha^5 = \alpha^4 \alpha = (\alpha^2 + \alpha + 1)\alpha = \alpha^3 + \alpha^2 + \alpha = \alpha^2 + 1 + \alpha^2 + \alpha = \alpha + 1
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| </math>
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| :<math>
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| \alpha^6 = \alpha^5 \alpha = (\alpha + 1)\alpha = \alpha^2 + \alpha
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| </math>
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| Using Zech logarithms to compute α<sup>6</sup> + α<sup>3</sup>:
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| :<math>\alpha^6 + \alpha^3 = \alpha^{6 + Z(-3)} = \alpha^{6 + Z(4)} = \alpha^{6 + 6} = \alpha^{12} = \alpha^5</math>
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| and verifying it in the polynomial representation:
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| :<math>\alpha^6 + \alpha^3 = (\alpha^2 + \alpha) + (\alpha^2 + 1) = \alpha + 1 = \alpha^5</math>
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Zech's Logarithms}}
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| [[Category:Linear algebra]]
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| [[Category:Finite fields]]
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