Some algorithms require a quadratic nonresidue in a given field. The first time a quadratic nonresidue is requested, one is generated at random, using the same source of random bits as other PBC random functions. [Which reminds me, should I get rid of the nqr field and instead have it as part of the data field in struct field_s?]

In fieldquadratic.c, a quadratic field extension is constructed with a square root of this randomly generated quadratic nonresidue in the base field. Thus for a nondeterminstic source of random bits, the same field may be constructed differently on different runs.

To construct the same field the same way every time, one must record the quadratic nonresidue generated from one run, and call field_set_nqr() every time this particular construction of a quadratic field extension is desired. Another use for this function is to save time by setting the quadratic nonresidue to some precomputed value.

Similarly, for higher degree extensions, a random irreducible polynomial may be chosen to construct it, but this must be recorded if the same construction is later required.

This happens behind the scenes in PBC.