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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