Bernstein seems to be correct that NIST did something dumb in calculating the time needed to break this algorithm. Basically, they said each iteration requires this expensive giant array access which takes about the time needed for 2^35 bit operations and that each iteration requires 2^25 bit operations. However, rather than adding the cost of the memory access to the cost of the bit operations in each iteration they multiplied them. That's bad [1].
But then Bernstein has to imply that this isn't just your usual bureaucratic stupidity but that it's somehow an attempt by the NSA to weaken our encryption standards. Yes, of course they consulted with the NSA because that's what they should do and it's part of the NSA's job to protect the security of US information. Any algorithm that NIST approves is going to be used by all sorts of government agencies and government contractors and if we'd be horribly unsafe if that algorithm could be cracked. That's presumably why the NSA helped with the design of the S-boxes in DES to make them more resistant to differential cryptoanalysis before the rest of the world knew about that.
This isn't even the *kind* of error that would be beneficial to the NSA. We aren't going to trick the Chinese or Russians into using an algorithm by multiplying numbers that should have been added. Even if the NSA is trying to make the new algorithm breakable they'd want to inject a flaw that only someone with some secret knowledge they have could break not just encourage NIST to adopt an algorithm where the best *public* attack takes less operations than they say it does.
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[1]: Though it's still possible that the issue is more subtle than this and the claim is that each operation requires one of these expensive array accesses but I can't find the source for these so can't check. That doesn't seem quite right but it might be how this thing got confused.