[ardacinar]

What about I randomly generate a number, add it to the illegal number to get a result. Is the resulting number illegal or not? (Thinking about it mathematically, the resulting number's distribution isn't really affected by the illegal number - Maybe we can practically find something by taking a lot of generated numbers and analyzing the apparent distribution)

Another idea is to generate a random number and multiply it with an illegal prime. If the illegal prime is a sufficiently large number; we can extract the original illegal prime with very high confidence by just finding its prime factors and picking the shortest one.

[droffel]

If you distribute a derivative of an illegal number, that derivative has the color of that illegal number. Whether anyone notices is a different question. The moment one of the intermediate steps involved the illegal number, the following steps gained it's color.