Crack rsa time

That, in turn, assumes that whatever software originally generated the RSA keys has the ability to do this. It's easy to find cases where things that ought to be random are relatively predictable; I remember gambling and adventure games on the old Apple ][ computer could be extremely predictable. Just after turning on the computer and starting a game, the first hand of Blackjack or the first storm at sea always came out exactly the same.

In fact, we have just seen another more modern example of this: the random numbers generated by gmpy in the "how many bit primes?

If we used that program to generate RSA keys, the same keys would be generated every time too. If random number generators are failing to produce truly unpredictable numbers, this can produce serious weaknesses in cryptography, because an attacker may have various ways to guess "secret" key values, or at least narrow down the possibilities dramatically.

Most computer systems today generate random numbers not primarily by measuring a physical quantity like radio static or lava lamp patterns but rather by using some sort of formula that gets fed with some ideally unpredictable value called a "seed". To get truly unpredictable numbers, we need truly unpredictable seeds from a large enough pool of possibilities.

The phenomenon where gmpy generates the same random numbers every time is an extreme case, where we haven't even attempted to give the random number generator a seed and it hasn't tried to create one from its environment. Security problems with inadequately or inappropriately sseded random number generators have arisen before; in , Ian Goldberg and David Wagner demonstrated a practical way of attacking the SSL implementation in Netscape Navigator because of predictable random-number generators.

In , the Debian Project revealed that one of its developers had mistakenly removed code from OpenSSL in that catastrophically weakened the random number generator causing RSA keys generated on Debian systems during that period to be limited to a pool of just 32, possibilities per key size! The idea that poor random number generators would make a collection of RSA keys jointly vulnerable to gcd, even though no individual key appears vulnerable in isolation, had been published as early as as a critique of RSA, but perhaps not experimentally demonstrated.

As Lenstra's group published its results, another group of researchers led by Nadia Heninger was simultaneously exploring the RSA public keys used on the Internet, and came up with similar results — likewise using a greatest-common-divisor method to find common factors.

Heninger's group found a more specific explanation for how the weak keys came to be; they were able to track down specific devices that have been habitually generating them! The devices in question "include[d] firewalls, routers, VPN devices, remote server administration devices, printers, projectors, and VOIP phones" from "more than thirty different manufacturers". The manufacturers in question were advised about the problem. These devices don't have keyboards and mice that they can use to help seed their random number generators as PCs often do, looking at the precise, unpredictable timing of user input , and they might be programmed to generate RSA keys automatically the first time they're turned on, perhaps without seeding the random number generator properly, or without having much with which to seed it.

Random number generators can be securely seeded with other kinds of hardware, too, but many device makers haven't realized this or haven't made provisions for taking advantage of the randomness that occurs in certain kinds of circuits.

Without proper seeding, the number of different primes they can conceivably generate is greatly reduced, and the chance that two devices will independently generate the same prime goes way up — an effect that eventually becomes visible when a large number of public keys are collected and compared. I mentioned above that calculating gcd is very fast. One useful way to calculate gcd — even for huge numbers — is with Euclid's algorithm , which Donald Knuth calls "the oldest nontrivial algorithm that has survived to the present day".

Euclid considered the problem of finding the gcd of two integers in Elements , book VII, proposition 2, and gave a very practical way to do it:. Let AB and CD be the two given numbers which are not prime to one another. So it is required to find the greatest common measure of AB and CD. And it is manifest that it is also the greatest common measure. For nothing greater than CD can measure CD. But if CD does not measure AB then some number will remain from AB and CD , the lesser being continually subtracted, in turn, from the greater, which will measure the number preceding it.

For a unit will not be left. The very opposite thing was assumed. Thus, some number will remain which will measure the number preceding it. Wikipedia calls this "the principle that the greatest common divisor of two numbers does not change if the smaller number is subtracted from the larger number".

Can you prove this, or follow Euclid's original proof in Fitzpatrick's edition at p. So a simple way of stating the subtraction-based form of Euclid's algorithm is: to find the greatest common divisor of two numbers, replace the larger number with the smaller number subtracted from the larger. Repeat this until one of the numbers reaches 1 in which case the gcd is 1, and the numbers are relatively prime or until one of the numbers reaches 0 in which case the gcd is the remaining value of the other number.

This is pretty fast, but there is an even faster way : instead of repeated subtraction, we can use the modulo operator to see what the result of the repeated subtractions will be. So the modulo-based form is: to find the greatest common divisor of two numbers, replace the larger number with the larger number mod the smaller. I note in passing that even though Euclid's gcd is incredibly fast compared to trying to factor numbers, it would still take quite a long time to apply to to every pair of RSA public keys on the whole Internet—since there are millions of them and the number of possible pairs of things grows with the square of the number of things.

There are still further tricks to make this process even faster in this case; Lenstra's team thought it best not to elaborate, saying " sapienti sat ", while Heninger's team credits techniques published in by Daniel J. For purposes of the present puzzle, the Euclidean gcd technique is totally sufficient because the number of public keys is reasonably small just one hundred.

Now, you can download the challenge file , which contains RSA public keys and corresponding messages, each message encrypted with one of the public keys.

The keys are bit, so we would normally not expect to be able to crack any of them. Expert opinion holds that some teams and organizations can do it with their resources and that keys of this size should no longer be used for security purposes.

However, some fraction of these keys are weak in the same way as the weak keys analyzed by Lenstra's and Heninger's groups: they share a prime with another key in our sample! Not all of the keys are weak, and I won't tell you ahead of time exactly how many are weak.

This will give you an estimate for brute force searching. You could also use the running time estimation for the quadratic sieve or the general number field sieve. This will give you estimates for factoring algorithms actually used by people breaking RSA numbers. Firstly, let's have a look at the size of numbers you're talking about. That's the size of number you're dealing with here, i. Next big question then is how is n constructed? The question then becomes how do we determine a number is prime and can we count them?

However, we can improve on that. You ought to realise fairly quickly that for any number, it is either prime or not. If it is not prime, then the gcd of it and at least one prime must be greater than 1 otherwise it would be prime. We conclude from this that any non-prime integer must be divisible by a set of primes.

A formal mathematical proof is not actually that much of a leap from here. This is called the fundamental theorem of arithemtic and simplifies matters slightly. So now, when working out if a number is prime, we no longer need to try every number, just the numbers we already know are prime!

This is clearly still really slow, so let's make another observation - given factors occur in pairs, the lower of the two numbers is at most the square root of the number.

If we are restricted to N the set of natural numbers it represents an upper bound on the largest possible value we need to check. So now, for any number N, we must search each integer starting from 2 and heading to sqrt N for each number we determine as prime in that list. We can then, if we find a prime, deduce whether it factors N itself.

I'm not going to estimate the run time of that because I'll undoubtedly say the wrong thing, but it'll take a long time. Now you see the strength of RSA. 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 coherence time at the moment is typically between microseconds, so you can forget about any calculation that takes a while!

Well, the problem is that innovation always comes in waves and sometimes breakthroughs are exactly that: they break through the established prediction. With the massive amount of research going into this field, it is hard to keep track of all the efforts. So we went from 1 billion qubits to 20m qubits in the space of 7 years. 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.