Threshold cryptosystem

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:

$\frac{1}{x(x-1)(x-2)},$

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

$\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-2},$

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:

$A(x-1)(x-2)+Bx(x-2)+Cx(x-1)=1,$

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

$(A+B+C)x^2-(3A+2B+C)x+2A=1.$

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:

$A+B+C=0,$

$3A+2B+C=0,$

$2A=1.$

Solving it results in:

$A=\frac{1}{2},B=-1,C=\frac{1}{2}.$

References

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