Comment by ggm
12 hours ago
The cost per bit is a doubling in time. So factoring a 512 RSA, compared to a 1024 RSA is significantly cheaper. The OP used contemporary hardware to do this. so, we'd have to ask if the orders of magnitude improvement in tech (QC aside) would permit 1024 in tractable time. I tend to no, but I appreciate there are other points of view. And of course, the belief that one day we can apply Shor with success exists. At which point the question is moot. Not that Shor does not itself demand significantly more stable gates, per extra bit of RSA. I always wonder why people don't look at the trend line in stable QuBits and the trendline in cost of RSA. Do the lines intersect?
Remember, Shor is like a coded gate level algorithm expressed as sequences of interconnected stable QuBits. So, if you double the cost for each RSA bit you add, its not "nothing" in terms of how you wire the rig.
(not a cryptographer, or a QC person so I expect to be hit by a very cold but stable quantum clue-by-four shortly. Maybe they have to hit me 1 million times, to confirm I'm hit. Its statistics.)
It’s not quite a doubling per bit, which is why RSA keys are relatively large compared to similar-strength ECDSA keys, for example.
Steve Weis, who has been doing RSA factoring on some large GPU clusters, estimates factoring 1024-bit RSA would take about 2000 GPU-years, which is well within the range of anyone with a serious budget.
Yeah, 2000 years sounds like a lot till you do the math. Apparently astra was trained on 100k Blackwell gpu’s. So just over 7 days to crack 1028-bit rsa on that cluster…
out of curiosity, how long would 2048-bit RSA take to factor?
It's hard to extrapolate that far, but maybe hundreds of thousands or millions of years.
Naively looking at scaling factors is going to be tricky, because computation of this scale is going to involve things like "how do I hijack every GPU on the planet", or worrying about when the sun will run out of hydrogen if you're using a single CPU.
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My article:
2048 Bit RSA and the Year 2030 https://articles.59.ca/doku.php?id=em:20482030
We don't have any way to predict when and if 2048 bit RSA would be factorable at this time. We would need a breakthrough in hardware and/or algorithms. The common estimation that it is equivalent to the difficulty of brute forcing symmetrical 112 bit encryption seems to be based on some sort of straightforward extrapolation. It doesn't take into account the amount of memory required for the poorly reducible matrix reduction step in the currently known best algorithm. That's 10^18 bytes of memory, or a million terabytes, somehow coupled to enough processing power to actually make anything possible.
Even if you accept the 112 bit estimate, that works out to something like 400 thousand years using the Bitcoin network as a reference to what we could reasonably achieve.
https://en.wikipedia.org/wiki/Key_size#Asymmetric_algorithm_...
says approximately the same as a 112-bit symmetric key, so 1/65536 as fast as however your target platform does at AES128, but probably 2000 times slower again because RSA is a really slow algorithm.
128-bit security is the de-facto minimum standard. Anything less than that is suspect. That's a 3072-bit RSA key. We only ever tolerated shorter keys because RSA is so slow. You should switch to ed25519 if you can.
2048-bit RSA gives something like 28 more bits of security than 1024-bit RSA has, so it would take about 250 million times as long to factor one 2048-bit key.
Doubling per bit is for symmetric encryption, where no attack better than brute force is known. RSA can be attacked using much faster techniques than brute force.
Can you point at some papers or articles that talk about attacks specifically on RSA? I've done a search and have a few references, but I'd be interested to know if you have any particular examples in mind.
I know that factoring (which attacks RSA) is sub-exponential, and I know that implementations of RSA (bad choices of primes, timing attacks, etc) can have weaknesses ... I'm just interested as to whether you have something else in mind.
Thx.
I think you're looking for the large formula at the top here: https://en.wikipedia.org/wiki/General_number_field_sieve
Reference to a scientific paper is given: https://www.ams.org/notices/199612/pomerance.pdf
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There are techniques to speed up the search for RSA keys quite significantly: they don't scale as with a pure brute force search, nor with a very useful rule of thumb (it's not even the case that doubling the RSA key length doubles its effective security, it's actually a fair bit less than that).