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Given current technology such as GPUs and GPU cracking software I was wondering if anyone has an idea on how long it would take to brute force the key used to derive an HMAC?

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If it's a proper key (128 random bits) then it's completely infeasible. If the key is derived from a password, it depends on the password and the derivation function. But in that case you don't attack HMAC, you attack the password. – CodesInChaos Mar 19 '13 at 21:38
up vote 6 down vote accepted

CodeInChaos has it right about the infeasbility of this against a random key; however, lets run the numbers to see how extremely correct he is:

Let us assume we are attacking HMAC-MD5 within TLS; this has a 128 bit key.

The fastest GPU server (actually, it has 25 GPUs internally) can test about 400 billion keys per second.

Let us assume that we, having a huge budget, have assembled 1 million of the above units; that means that we can test $4 \times 10^{17}$ keys per second; sounds impressive.

However, it'll take an average of $2^{127}$ keys tested before we recover the HMAC-MD5 key; at $4 \times 10^{17}$ keys per second, that'd be $4 \times 10^{20}$ seconds, or about $10,000,000,000$ years. I'd call that somewhat impractical; if nothing else, the sun will become a red giant and destroy our set up before we're likely to find a result.

It gets even worse if we talk about HMAC-SHA1; that uses 160 bit keys; that increases the amount of time we'd expect to take by about 4 billion.

Yes, if what we're talking about is a key derived from a password, GPUs can be used. However, when we're talking about random keys, they don't come close to scratching the algorithm.

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