Generalized Chinese Remainder Theorem


A second take at the Chinese Remainder Theorem, after using the strict version in Advent of Code 2020 - Day 13.

That previous post introduced a small implementation of the gist of the Chinese Remainder Theorem, namely:

sub crt ($n1, $r1, $n2, $r2) {
   my ($gcd, $x, $y) = egcd($n1, $n2);
   die "not coprime! <$n1> <$n2>" if $gcd != 1;
   my $N = $n1 * $n2;
   my $r = ($r2 * $x * $n1 + $r1 * $y * $n2) % $N;
   return [$N, $r];

That was good for that day’s puzzle - after all, the input I was given (well, this was valid for everyone else too) indeed had only coprime values for the bus numbers, so the die line was actually not needed at all (it’s there to make sure of that and leaving it does not harm anyway, acting as a sort of reminder).

But that’s not the whole story.

And I knew it.

And it came to haunt me just like the Ghost of Christmas Yet to Come.

It’s indeed possible to apply the theorem, or better a slight generalization of it, also to cases when the coprimeness (is this a word? 🧐) requirement is failed, namely (adapting from Chinese Remainder Theorem):

If $r_1 \equiv r_2 \pmod{gcd(n_1, n_2)}$ then this system of equations has a unique solution modulo $\frac{n_1 \cdot n_2}{gcd(n_1, n_2)}$. Otherwise, it has no solutions.

Honestly, as it often happens, it both makes a lot of sense and it’s not very scaring to implement at all. So I decided to adapt the implementation with this generalization, and also to accept a generic number of pairs of modulus and remainder, resulting in this:

sub chinese_remainder_theorem {
   die "no inputs" unless scalar @_;
   die "need an even number of inputs" if scalar(@_) % 2 == 1;
   my ($N, $R) = splice @_, 0, 2;
   while (@_) {
      my ($n, $r) = splice @_, 0, 2;
      my ($gcd, $x, $y) = egcd($N, $n);
      if ($gcd != 1) {
         die "cannot combine: {x ~ $R (mod $N)} with {$x ~ $r (mod $n)}"
            unless ($R % $gcd) == ($r % $gcd);
         $_ /= $gcd for ($N, $n);
      my $P = $N * $n;
      ($N, $R) = ($P, ($r * $x * $N + $R * $y * $n) % $P);
   return ($N, $R);

As there are a lot of multiplications, integers can explode very fast and yield wrong results due to representation issues. For this reason, you might want to use the big integers version instead:

sub chinese_remainder_theorem_bi {
   require Math::BigInt;
   return chinese_remainder_theorem(map { Math::BigInt->new($_) } @_);

It’s just a thin wrapper around the other function, making sure to feed it with Math::BigInt objects instead of plain Perl integers, so that nothing will be lost on the way.

Well, some time might be lost, probably. But it’s for the right cause of correctness 😅.

Both functions ended up in cglib of course, because… why not? There might be a puzzle in the future that can use some copy-and-paste love 🙄

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