Excludent

A method which can be used to solve any Quadratic Congruence. This technique relies on the fact that solving

$$x^2\equiv b\ \left({{\rm mod\ } {p}}\right)$$

is equivalent to finding a value $y$ such that

$$b+py=x^2.$$

Pick a few small moduli $m$. If $y$ mod $m$ does not make $b+py$ a quadratic residue of $m$, then this value of $y$ may be excluded. Furthermore, values of $y> p/4$ are never necessary.