A necessary and sufficient condition for the asymptotic idealness of the GRS threshold secret sharing scheme

The study of the asymptotic idealness of the Goldreich-Ron-Sudan (GRS, for short) threshold secret sharing scheme was the subject of several research papers, where sufficient conditions were provided. In this paper a necessary and sufficient condition is established; namely, it is shown that the GRS threshold secret sharing scheme is asymptotically ideal under the uniform distribution on the secret space if and only if it is based on 1-compact sequences of co-primes.