A new record has been set for the largest encryption key ever cracked — but your secrets should be safe for now. Long strings of numbers are essential to the encryption that keeps our online data safe. One widely used form of encryption called RSA cryptography relies on the fact that it is extremely difficult to find the prime numbers that multiply together to yield very large numbers.
The inventors of the RSA algorithm published a list of RSA keys and challenged people to find the original primes, as a way of tracking how secure the encryption is against modern computers.
The previous RSA record was set in , with a key of decimal digits and bits. Pick a very large prime and you end up with a long way to go. As it currently stands, we have to start from 2, which is clearly awful. Primality testing aims to improve on that using a variety of techniques. The naive method is the one we've just discussed. I think a detailed discussion of these techniques is probably more appropriate for Math, so let me sum it up: all of the runtimes are rubbish and using this as a way to count primes would be horrendous.
So, we cannot count the number of primes reliably less than a number without taking forever, since it's effectively analogous to integer factorisation. What about a function that somehow counts primes some other way? It is, however, exactly that; the aim of such a function is to exactly count the number of primes but at present it simply gives you an estimate. For your purposes, this could be considered good enough. However, it is absolutely still an approximation.
Take a look at the rest of the article. Amongst other things, other estimations are dependent on the Riemann Hypothesis. Ok, so, what about integer factorisation? Well, the second best method to date is called the Quadratic Sieve and the best is called the general number field sieve. Both of these methods touch some fairly advanced maths; assuming you're serious about factoring primes I'd get reading up on these. Certainly you should be able to use the estimates for both as improvements to using the prime number theorem, since if you're going to factor large primes, you want to be using these and not a brute force search.
Ok, fair enough. Integer factorisation on a quantum computer can be done in ridiculously short amounts of time assuming we will be able to implement Shor's Algorithm. I should point out, however, this requires a quantum computer. As far as I'm aware, the development of quantum computers of the scale that can crack RSA is currently a way off. See quantum computing developments. In any case, Shor's Algorithm would be exponentially faster. The page on it gives you an estimate for the run time of that, which you may like to include in your estimates.
Another option is to create a big database of possible keys and use it as a lookup table. Apparently you don't even need ALL the primes, just a couple will get you through a big percentage of internet traffic. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
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I didn't want to claim that the bit factoring thing was bogus since I know far less about quantum or the whole area in general than you do, but I had a strong feeling it was. I also had a fair idea factoring algorithms were tailored per number, but not to the extent you describe, as my number theory just isn't that good yet. I haven't been around this site long, but I'm getting really good at guessing which questions are going to be answered by that bear.
Some years later, bits is down to "about 7. The sort of organizations who might possibly be don't make public announcements: crypto. In , researchers successfully factored a bit integer basically breaking RSA They had to use many hundreds of machines over a timeframe of 2 years!
So, what is the biggest number that has been factored by a quantum computer available today? Quantum Annealing is emerging as a powerful force to be reckoned with. Quantum Annealing Devices e. They can only solve special optimization problems, but because of these limitations, they have been around for a longer time, are much more mature and have many more qubits than universal quantum computers.
As it turns out, the problem of factoring integers can also be formulated as an optimization problem. They were able to factor the number , in with qubits. That is already an bit number. That is now already a bit number.
These time estimates assume that no fundamental breakthrough from an algorithmic side will be made and the same algorithm will be run on a D-Wave device just with more qubits. This is obviously a massive simplification as there is so much research happening at the moment, which will inevitably lead to breakthroughs.
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