Rational reconstruction (mathematics)

In mathematics, rational reconstruction is a method that allows one to recover a rational number from its value modulo an integer. If a problem with a rational solution ${\frac {r}{s}}$ is considered modulo a number m, one will obtain the number $n=r\times s^{-1}{\pmod {m}}$. If |r| < N and 0 < s < D then r and s can be uniquely determined from n if m > 2ND using the Euclidean algorithm, as follows.

One puts $v=(m,0)$ and $w=(n,1)$. One then repeats the following steps until the first component of w becomes $\leq N$. Put $q=\left\lfloor {\frac {v_{1}}{w_{1}}}\right\rfloor$, put z = v − qw. The new v and w are then obtained by putting v = w and w = z.

Then with w such that $w_{1}\leq N$, one makes the second component positive by putting w = −w if $w_{2}<0$. If $w_{2}<D$ and $\gcd(w_{1},w_{2})=1$, then the fraction ${\frac {r}{s}}$ exists and $r=w_{1}$ and $s=w_{2}$, else no such fraction exists.