How Far Ahead of Academia Are Government Agencies?
Disclaimer: This post is possibly opinion based.

How far ahead (if at all) are governmental agencies of open source (specifically academic) research?

This question is impossible to answer. By definition, this require a knowledge of the state of research in such agencies which is definitively something they want to be kept as secret...
thx to @fkraiem:

However one can get a feeling of it by considering the followings:

Academic research is usually founded by companies or government agencies.
e.g. for the Gimli paper:

This work was supported in part by the Commission of the European Communities through the Horizon 2020 program under project number 645622 (PQCRYPTO) and project number 645421 (ECRYPT-CSA); the Austrian Science Fund (FWF) under grant P26494-N15; the ARC project NANOSEC; the Belgian Fund for Scientific Research (FNRS-F.R.S.); the Technology Foundation STW (project 13499 TYPHOON), from the Dutch government; the Netherlands Organisation for Scientific Research (NWO) under grant 639.073.005; and the U.S. National Science Foundation under grant 1314919. “Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.


In the case of important projects, academic research is paired with companies with huge computing power (Google, Microsoft Research...) e.g. SHAttered:

This result is the product of a long term collaboration between the Cryptology Group at Centrum Wiskunde &amp; Informatica (CWI) - the national research institute for mathematics and computer science in the Netherlands - and the Google Research Security, Privacy and Anti-abuse Group. Two years ago Marc Stevens and Elie Bursztein, who leads the Google's anti-abuse research team, began collaborating on making Marc's cryptanalytic attacks against SHA-1 practical by leveraging Google expertise and infrastructure.

Sure such companies may? still not have the computing power of NSA, but the minds in there are definitively on par with such agencies.


I would also advise this to have a look at this: The Moral Character of Cryptographic Work by Phillip Rogaway [paper] [Usenix Talk], specially at the founding part (p. 36). We see a increase in the number of project founded by military agencies.
WARNING: this is a personal assumption, you may chose to disagree with it.
Knowing their missions (cf quote bellow) we can somewhat deduce that their level of research is most likely on par with academic work (why would you heavily fund something that would be bellow your current level of comprehension?)
e.g. for Defense Advanced Research Projects Agency (DARPA) :

The genesis of that mission and of DARPA itself dates to the launch of Sputnik in 1957, and a commitment by the United States that, from that time forward, it would be the initiator and not the victim of strategic technological surprises.
source.

Their goal are the following:

fund academic research
but most importantly know what you are currently researching on! $\Leftarrow$ This is information they want!


TL;DR:  I believe they are on par with current academic research.