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Is there any new understanding, property, or knowledge that has come from the Number Fields @Home distributed computing project? Has any outcome advanced the study of cryptography, or altered previous recommendations?
If nothing in particular has come from the project as of yet, are there some outcomes a cryptographer would find intriguing? If so, what are they, and in what way?
TL;DR: The NumberFields@home Project has not benefited cryptography.
Number fields play an important role in cryptography, especially when talking about the RSA cryptosystem.
The security of RSA is based on the presumtion that factoring a large semi-prime (the product of two prime numbers) is very hard to achieve, this problem is known as the integer factorization problem.
The Quadratic sieve is an algorithm for factorization. It's the fastest known algorithm for factoring a number $n$, where $n < 10^{100}$.
The General number field sieve is the fastest known algorithm for factoring a number $n$, where $n > 10^{100}$.
It currently holds the record for factoring the RSA-number RSA-768, meaning it has factored a 768-bit semi-prime. A detailed describtion about how this was achieved can be found in this published paper.
Well... Less so. Factoring a 1024-bit RSA modulus would be about a thousand times harder than a 768-bit. They have achieved factoring this number by a collection of parallel computers and the total time amounted approximately to the equivalent of almost 2000 years of computing on a single-core 2.2 GHz AMD Opteron-based computer.
Quoting the previously mentioned paper from February 18, 2010:
Because the first factorization of a 512-bit RSA modulus was reported only a decade ago it is not unreasonable to expect that 1024-bit RSA moduli can be factored well within the next decade by an academic effort such as ours or the one in. Thus, it would be prudent to phase out usage of 1024-bit RSA within the next three to four years.
But you can rest assured that the current standard (2048-bit RSA modulus) is safe.
Number Fields @ home has yet to reveal some "breakthrough" for cryptography. You can see a list of there achievements here, so strictly speaking the answer is No.
But it might reveal something someday, even though the purpose of this project is "backwards", since the goal of this project is not to prove anything directly, but rather to gather information that might answer a question we maybe haven't asked yet.
To quote the project:
The project as a whole is basic research, in effect, charting unknown territory. In the future, this may have a bearing on a number of questions.
And quoting directly the impact on cryptography:
Number fields are used in some modern factoring algorithms which are relevant to attacks on RSA. Other researchers have investigated using properties of number fields as the basis of new cryptosystems. It is not clear what number fields will be useful in this research, but the more we know about the general landscape of number fields, the better.
