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. ^[1]

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.

References

1. P. S. Wang, a p-adic algorithm for univariate partial fractions, Proceedings of SYMSAC ´81, ACM Press, 212 (1981); P. S. Wang, M. J. T. Guy, and J. H. Davenport, p-adic reconstruction of rational numbers, SIGSAM Bulletin 16 (1982).