Threshold cryptosystem: Difference between revisions

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In mathematics, the method of '''equating the coefficients''' is a way of solving a functional equation of two [[polynomial]]s for a number of unknown parameters. It relies on the fact that two polynomials are identical precisely when all corresponding coefficients are equal. The method is used to bring formulas into a desired form.
 
==Example==
Suppose we want to apply [[partial fraction decomposition]] to the expression:
 
:<math>\frac{1}{x(x-1)(x-2)},\,</math>
 
that is, we want to bring it into the form:
 
:<math>\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-2},\,</math>
 
in which the unknown parameters are ''A'', ''B'' and ''C''.
Multiplying these formulas by ''x''(''x''&nbsp;&minus;&nbsp;1)(''x''&nbsp;&minus;&nbsp;2) turns both into polynomials, which we equate:
 
:<math>A(x-1)(x-2) + Bx(x-2) + Cx(x-1) = 1,\,</math>
 
or, after expansion and collecting terms with equal powers of ''x'':
 
:<math>(A+B+C)x^2 - (3A+2B+C)x + 2A = 1.\,</math>
 
At this point it is essential to realize that the polynomial 1 is in fact equal to the polynomial 0''x''<sup>2</sup>&nbsp;+&nbsp;0''x''&nbsp;+&nbsp;1, having zero coefficients for the positive powers of ''x''.  Equating the corresponding coefficients now results in this [[system of linear equations]]:
 
:<math>A+B+C = 0,\,</math>
:<math>3A+2B+C = 0,\,</math>
:<math>2A = 1.\,</math>
 
Solving it results in:
 
:<math>A = \frac{1}{2},\, B = -1,\, C = \frac{1}{2}.\,</math>
 
==References==
*{{cite book |title=Encyclopedia of Mathematics |first=James |last=Tanton |page=162 |publisher=Facts on File |year=2005 |isbn=0-8160-5124-0}}
 
{{DEFAULTSORT:Equating Coefficients}}
[[Category:Elementary algebra]]
[[Category:Equations]]

Latest revision as of 11:45, 11 November 2013

In mathematics, the method of equating the coefficients is a way of solving a functional equation of two polynomials for a number of unknown parameters. It relies on the fact that two polynomials are identical precisely when all corresponding coefficients are equal. The method is used to bring formulas into a desired form.

Example

Suppose we want to apply partial fraction decomposition to the expression:

that is, we want to bring it into the form:

in which the unknown parameters are A, B and C. Multiplying these formulas by x(x − 1)(x − 2) turns both into polynomials, which we equate:

or, after expansion and collecting terms with equal powers of x:

At this point it is essential to realize that the polynomial 1 is in fact equal to the polynomial 0x2 + 0x + 1, having zero coefficients for the positive powers of x. Equating the corresponding coefficients now results in this system of linear equations:

Solving it results in:

References

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