Let n/d be in Q, m be a positive integer and let u = n/d mod m.
Thus u is the image of a rational number modulo m.
The rational reconstruction problem is; given u and m find n/d.
A solution was first given by Wang in 1981.
Wang's algorithm outputs n/d when m > 2 M^2 where M = max(n,d).
Because of the wide application of this algorithm in computer algebra,
several authors have investigated its practical efficiency
and asymptotic time complexity.
In this paper we present a new solution which is almost optimal in
the following sense; with controllable high probability, our algorithm
will output n/d when m is only a few bits longer than 2 n d.
Further, our algorithm will fail with high probability when
m < 2 n d. This means that in a modular algorithm where m is a
product of primes, the modular algorithm will need about one prime
more than the minimum necessary to reconstruct n/d; thus if
n << d or d << n the new algorithm saves up to half the
number of primes.
